Series ISSN: 1938-1743 ANEASY PATH AND APPLICATIONS TOCONVEX ANALYSIS • NAM MORDUKHOVICH
SYNTHESIS LECTURES ON M Morgan& Claypool Publishers COMPUTER MATHEMATICS AND STATISTICS &C
Series Editor: Steven G. Krantz, Washington University, St. Louis An Easy Path to Convex Analysis and An Easy Path to Convex Applications Analysis and Applications Boris S. Mordukhovich, Wayne State University Nguyen Mau Nam, Portland State University
Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by researchers of different fields. As convex analysis is the mathematical foundation for convex optimization, having deep knowledge of convex analysis helps students and researchers apply its tools more effectively. The main goal of this book is to provide an easy access to the most fundamental parts of convex analysis and its applications to optimization. Modern techniques of variational analysis are employed to clarify and simplify some basic proofs in convex analysis and build the theory of generalized differentiation for convex functions and sets in finite dimensions. We also present new applications of convex analysis to location problems in connection with many interesting geometric problems such as the Fermat-Torricelli problem, the Heron problem, the Sylvester problem, and their generalizations. Of course, we do not expect to touch every aspect of convex analysis, but the book consists of sufficient material for a first course on this subject. It can also serve as supplemental reading material for a Boris S. Mordukhovich course on convex optimization and applications. Nguyen Mau Nam
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An Easy Path to Convex Analysis and Applications Synthesis Lectures on Mathematics and Statistics
Editor Steven G. Krantz, Washington University, St. Louis An Easy Path to Convex Analysis and Applications Boris S. Mordukhovich and Nguyen Mau Nam 2014 Applications of Affine and Weyl Geometry Eduardo García-Río, Peter Gilkey, Stana Nikčević, and Ramón Vázquez-Lorenzo 2013 Essentials of Applied Mathematics for Engineers and Scientists, Second Edition Robert G. Watts 2012 Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations Goong Chen and Yu Huang 2011 Matrices in Engineering Problems Marvin J. Tobias 2011 e Integral: A Crux for Analysis Steven G. Krantz 2011 Statistics is Easy! Second Edition Dennis Shasha and Manda Wilson 2010 Lectures on Financial Mathematics: Discrete Asset Pricing Greg Anderson and Alec N. Kercheval 2010 iii Jordan Canonical Form: eory and Practice Steven H. Weintraub 2009 e Geometry of Walker Manifolds Miguel Brozos-Vázquez, Eduardo García-Río, Peter Gilkey, Stana Nikčević, and Ramón Vázquez-Lorenzo 2009 An Introduction to Multivariable Mathematics Leon Simon 2008 Jordan Canonical Form: Application to Differential Equations Steven H. Weintraub 2008 Statistics is Easy! Dennis Shasha and Manda Wilson 2008 A Gyrovector Space Approach to Hyperbolic Geometry Abraham Albert Ungar 2008 Copyright © 2014 by Morgan & Claypool
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An Easy Path to Convex Analysis and Applications Boris S. Mordukhovich and Nguyen Mau Nam www.morganclaypool.com
ISBN: 9781627052375 paperback ISBN: 9781627052382 ebook
DOI 10.2200/S00554ED1V01Y201312MAS014
A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS
Lecture #14 Series Editor: Steven G. Krantz, Washington University, St. Louis Series ISSN Synthesis Lectures on Mathematics and Statistics Print 1938-1743 Electronic 1938-1751 An Easy Path to Convex Analysis and Applications
Boris S. Mordukhovich Wayne State University
Nguyen Mau Nam Portland State University
SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS #14
M &C Morgan& cLaypool publishers ABSTRACT Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by re- searchers of different fields. As convex analysis is the mathematical foundation for convex opti- mization, having deep knowledge of convex analysis helps students and researchers apply its tools more effectively. e main goal of this book is to provide an easy access to the most fundamental parts of convex analysis and its applications to optimization. Modern techniques of variational analysis are employed to clarify and simplify some basic proofs in convex analysis and build the theory of generalized differentiation for convex functions and sets in finite dimensions. We also present new applications of convex analysis to location problems in connection with many in- teresting geometric problems such as the Fermat-Torricelli problem, the Heron problem, the Sylvester problem, and their generalizations. Of course, we do not expect to touch every aspect of convex analysis, but the book consists of sufficient material for a first course on this subject. It can also serve as supplemental reading material for a course on convex optimization and applications.
KEYWORDS Affine set, Carathéodory theorem, convex function, convex set, directional deriva- tive, distance function, Fenchel conjugate, Fermat-Torricelli problem, generalized differentiation, Helly theorem, minimal time function, Nash equilibrium, normal cone, Radon theorem, optimal value function, optimization, smallest enclosing ball problem, set-valued mapping, subdifferential, subgradient, subgradient algorithm, support function, Weiszfeld algorithm vii
e first author dedicates this book to the loving memory of his father S M (1924–1993), a kind man and a brave warrior.
e second author dedicates this book to the memory of his parents.
ix Contents
Preface ...... xi
Acknowledgments ...... xiii
List of Symbols ...... xv
1 Convex Sets and Functions ...... 1 1.1 Preliminaries ...... 1 1.2 Convex Sets ...... 5 1.3 Convex Functions ...... 10 1.4 Relative Interiors of Convex Sets ...... 21 1.5 e Distance Function ...... 28 1.6 Exercises for Chapter 1 ...... 32 2 Subdifferential Calculus ...... 37 2.1 Convex Separation ...... 37 2.2 Normals to Convex Sets ...... 42 2.3 Lipschitz Continuity of Convex Functions ...... 49 2.4 Subgradients of Convex Functions ...... 53 2.5 Basic Calculus Rules ...... 61 2.6 Subgradients of Optimal Value Functions ...... 70 2.7 Subgradients of Support Functions ...... 75 2.8 Fenchel Conjugates ...... 77 2.9 Directional Derivatives ...... 82 2.10 Subgradients of Supremum Functions ...... 86 2.11 Exercises for Chapter 2 ...... 90 3 Remarkable Consequences of Convexity...... 95 3.1 Characterizations of Differentiability ...... 95 3.2 Carathéodory eorem and Farkas Lemma ...... 97 3.3 Radon eorem and Helly eorem ...... 102 x 3.4 Tangents to Convex Sets ...... 103 3.5 Mean Value eorems ...... 106 3.6 Horizon Cones ...... 108 3.7 Minimal Time Functions and Minkowski Gauge ...... 111 3.8 Subgradients of Minimal Time Functions ...... 117 3.9 Nash Equilibrium ...... 120 3.10 Exercises for Chapter 3 ...... 124 4 Applications to Optimization and Location Problems ...... 129 4.1 Lower Semicontinuity and Existence of Minimizers ...... 129 4.2 Optimality Conditions ...... 135 4.3 Subgradient Methods in Convex Optimization ...... 140 4.4 e Fermat-Torricelli Problem ...... 146 4.5 A Generalized Fermat-Torricelli Problem ...... 154 4.6 A Generalized Sylvester Problem ...... 168 4.7 Exercises for Chapter 4 ...... 178
Solutions and Hints for Exercises ...... 183
Bibliography ...... 195
Authors’ Biographies ...... 199
Index ...... 201 xi Preface
Some geometric properties of convex sets and, to a lesser extent, of convex functions had been studied before the 1960s by many outstanding mathematicians, first of all by Hermann Minkowski and Werner Fenchel. At the beginning of the 1960s convex analysis was greatly de- veloped in the works of R. Tyrrell Rockafellar and Jean-Jacques Moreau who initiated a systematic study of this new field. As a fundamental part of variational analysis, convex analysis contains a generalized differentiation theory that can be used to study a large class of mathematical mod- els with no differentiability assumptions imposed on their initial data. e importance of convex analysis for many applications in which convex optimization is the first to name has been well recognized. e presence of convexity makes it possible not only to comprehensively investigate qualitative properties of optimal solutions and derive necessary and sufficient conditions for opti- mality but also to develop effective numerical algorithms to solve convex optimization problems, even with nondifferentiable data. Convex analysis and optimization have an increasing impact on many areas of mathematics and applications including control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, statistics, economics and finance, etc. ere are several fundamental books devoted to different aspects of convex analysis and optimization. Among them we mention “Convex Analysis” by Rockafellar [26], “Convex Analysis and Minimization Algorithms” (in two volumes) by Hiriart-Urruty and Lemaréchal [8] and its abridge version [9], “Convex Analysis and Nonlinear Optimization” by Borwein and Lewis [4], “Introductory Lectures on Convex Optimization” by Nesterov [21], and “Convex Optimization” by Boyd and Vandenberghe [3] as well as other books listed in the bibliography below. In this big picture of convex analysis and optimization, our book serves as a bridge for students and researchers who have just started using convex analysis to reach deeper topics in the field. We give detailed proofs for most of the results presented in the book and also include many figures and exercises for better understanding the material. e powerful geometric approach developed in modern variational analysis is adopted and simplified in the convex case in order to provide the reader with an easy path to access generalized differentiation of convex objects in finite dimensions. In this way, the book also serves as a starting point for the interested reader to continue the study of nonconvex variational analysis and applications. It can be of interest from this viewpoint to experts in convex and variational analysis. Finally, the application part of this book not only concerns the classical topics of convex optimization related to optimality conditions and subgradient algorithms but also presents some recent while easily understandable qualitative and numerical results for important location problems. xii PREFACE e book consists of four chapters and is organized as follows. In Chapter 1 we study fundamental properties of convex sets and functions while paying particular attention to classes of convex functions important in optimization. Chapter 2 is mainly devoted to developing basic calculus rules for normals to convex sets and subgradients of convex functions that are in the mainstream of convex theory. Chapter 3 concerns some additional topics of convex analysis that are largely used in applications. Chapter 4 is fully devoted to applications of basic results of convex analysis to problems of convex optimization and selected location problems considered from both qualitative and numerical viewpoints. Finally, we present at the end of the book Solutions and Hints to selected exercises. Exercises are given at the end of each chapter while figures and examples are provided throughout the whole text. e list of references contains books and selected papers, which are closely related to the topics considered in the book and may be helpful to the reader for advanced studies of convex analysis, its applications, and further extensions. Since only elementary knowledge in linear algebra and basic calculus is required, this book can be used as a textbook for both undergraduate and graduate level courses in convex analysis and its applications. In fact, the authors have used these lecture notes for teaching such classes in their universities as well as while visiting some other schools. We hope that the book will make convex analysis more accessible to large groups of undergraduate and graduate students, researchers in different disciplines, and practitioners.
Boris S. Mordukhovich and Nguyen Mau Nam December 2013 xiii Acknowledgments
e first author acknowledges continued support by grants from the National Science Foundation and the second author acknowledges support by the Simons Foundation.
Boris S. Mordukhovich and Nguyen Mau Nam December 2013
xv List of Symbols
R the real numbers R . ; the extended real line D 1 1 R the nonnegative real numbers C R> the positive real numbers N the positive integers co ˝ convex hull of ˝ aff ˝ affine hull of ˝ int ˝ interior of ˝ ri ˝ relative interior of ˝ span ˝ linear subspace generated by ˝ cone ˝ cone generated by ˝ K˝ convex cone generated by ˝ dim ˝ dimension of ˝ ˝ closure of ˝ bd ˝ boundary of ˝ IB closed unit ball IB.x r/ closed ball with center x and radius r NI N dom f domain of f epi f epigraph of f gph F graph of mapping F LŒa; b line connecting a and b d.x ˝/ distance from x to ˝ I ˘.x ˝/ projection of x to ˝ I x; y inner product of x and y h i A adjoint/transpose of linear mapping/matrix A N.x ˝/ normal cone to ˝ at x NI N ker A kernel of linear mapping A D F.x; y/ coderivative to F at .x; y/ N N N N T.x ˝/ tangent cone to ˝ at x NI N F horizon/asymptotic cone of F T1F ˝ minimal time function defined by constant dynamic F and target set ˝ L.x; / Lagrangian F Minkowski gauge of F xvi LIST OF SYMBOLS
˘F . ˝/ generalized projection defined by the minimal time function I GF .x; t/ generalized ball defined by dynamic F with center x and radius t N N 1 C H A P T E R 1 Convex Sets and Functions
is chapter presents definitions, examples, and basic properties of convex sets and functions in the Euclidean space Rn and also contains some related material.
1.1 PRELIMINARIES We start with reviewing classical notions and properties of the Euclidean space Rn. e proofs of the results presented in this section can be found in standard books on advanced calculus and linear algebra. n n Let us denote by R the set of all n tuples of real numbers x .x1; : : : ; xn/. en R is D a linear space with the following operations:
.x1; : : : ; xn/ .y1; : : : ; yn/ .x1 y1; : : : ; xn yn/; C D C C .x1; : : : ; xn/ .x1; : : : ; xn/; D n n where .x1; : : : ; xn/; .y1; : : : ; yn/ R and R. e zero element of R and the number zero 2 2 of R are often denoted by the same notation 0 if no confusion arises. n T For any x .x1; : : : ; xn/ R , we identify it with the column vector x Œx1; : : : ; xn , D 2 D n where the symbol “T ” stands for vector transposition. Given x .x1; : : : ; xn/ R and y n D 2 D .y1; : : : ; yn/ R , the inner product of x and y is defined by 2 n
x; y X xi yi : h i WD i 1 D e following proposition lists some important properties of the inner product in Rn.
Proposition 1.1 For x; y; z Rn and R, we have: 2 2 (i) x; x 0, and x; x 0 if and only if x 0. h i h i D D (ii) x; y y; x . h i D h i (iii) x; y x; y . h i D h i (iv) x; y z x; y x; z . h C i D h i C h i n e Euclidean norm of x .x1; : : : ; xn/ R is defined by D 2 x qx2 ::: x2: k k WD 1 C C n 2 1. CONVEX SETS AND FUNCTIONS It follows directly from the definition that x p x; x . k k D h i Proposition 1.2 For any x; y Rn and R, we have: 2 2 (i) x 0, and x 0 if and only if x 0. k k k k D D (ii) x x . k k D j j k k (iii) x y x y .the triangle inequality/. k C k Ä k k C k k (iv) x; y x y .the Cauchy-Schwarz inequality/. jh ij Ä k k k k Using the Euclidean norm allows us to introduce the balls in Rn, which can be used to define other topological notions in Rn.
Definition 1.3 e centered at x with radius r 0 and the of Rn N are defined, respectively, by
IB.x r/ ˚x Rn ˇ x x r« and IB ˚x Rn ˇ x 1«: NI WD 2 ˇ k Nk Ä WD 2 ˇ k k Ä It is easy to see that IB IB.0 1/ and IB.x r/ x rIB. D I NI DN C Definition 1.4 Let ˝ Rn. en x is an of ˝ if there is ı > 0 such that N IB.x ı/ ˝: NI e set of all interior points of ˝ is denoted by int ˝. Moreover, ˝ is said to be if every point of ˝ is its interior point.
We get that ˝ is open if and only if for every x ˝ there is ı > 0 such that IB.x ı/ ˝. N 2 NI It is obvious that the empty set and the whole space Rn are open. Furthermore, any open ball ; B.x r/ x Rn x x < r centered at x with radius r is open. NI WD f 2 j k Nk g N Definition 1.5 A set ˝ Rn is if its complement ˝c Rn ˝ is open in Rn. D n It follows that the empty set, the whole space, and any ball IB.x r/ are closed in Rn. NI Proposition 1.6 (i) e union of any collection of open sets in Rn is open. (ii) e intersection of any finite collection of open sets in Rn is open. (iii) e intersection of any collection of closed sets in Rn is closed. (iv) e union of any finite collection of closed sets in Rn is closed.
n Definition 1.7 Let xk be a sequence in R . We say that xk to x if f g f g N xk x 0 as k . In this case we write k Nk ! ! 1
lim xk x: k DN !1 1.1. PRELIMINARIES 3 is notion allows us to define the following important topological concepts for sets.
Definition 1.8 Let ˝ be a nonempty subset of Rn. en: (i) e of ˝, denoted by ˝ or cl ˝, is the collection of limits of all convergent sequences be- longing to ˝. (ii) e of ˝, denoted by bd ˝, is the set ˝ int ˝. n We can see that the closure of ˝ is the intersection of all closed sets containing ˝ and that the interior of ˝ is the union of all open sets contained in ˝. It follows from the definition that x ˝ if and only if for any ı > 0 we have IB.x ı/ ˝ . Furthermore, x bd ˝ if and only N 2 NI \ ¤ ; N 2 if for any ı > 0 the closed ball IB.x ˝/ intersects both sets ˝ and its complement ˝c. NI n Definition 1.9 Let xk be a sequence in R and let k` be a strictly increasing sequence of positive f g f g integers. en the new sequence xk is called a of xk . f ` g f g We say that a set ˝ is bounded if it is contained in a ball centered at the origin with some radius r > 0, i.e., ˝ IB.0 r/. us a sequence xk is bounded if there is r > 0 with I f g
xk r for all k N : k k Ä 2 e following important result is known as the Bolzano-Weierstrass theorem.
eorem 1.10 Any bounded sequence in Rn contains a convergent subsequence.
e next concept plays a very significant role in analysis and optimization.
Definition 1.11 We say that a set ˝ is in Rn if every sequence in ˝ contains a subsequence converging to some point in ˝.
e following result is a consequence of the Bolzano-Weierstrass theorem.
eorem 1.12 A subset ˝ of Rn is compact if and only if it is closed and bounded.
n For subsets ˝, ˝1, and ˝2 of R and for R, we define the operations: 2
˝1 ˝2 ˚x y ˇ x ˝1; y ˝2«; ˝ ˚x ˇ x ˝«: C WD C ˇ 2 2 WD ˇ 2 e next proposition can be proved easily.
n Proposition 1.13 Let ˝1 and ˝2 be two subsets of R .
(i) If ˝1 is open or ˝2 is open, then ˝1 ˝2 is open. C (ii) If ˝1 is closed and ˝2 is compact, then ˝2 ˝2 is closed. C 4 1. CONVEX SETS AND FUNCTIONS Recall now the notions of bounds for subsets of the real line.
Definition 1.14 Let D be a subset of the real line. A number m R is a of D if we 2 have x m for all x D: 2 If the set D has a lower bound, then it is . Similarly, a number M R is an 2 of D if x M for all x D; Ä 2 and D is if it has an upper bound. Furthermore, we say that the set D is if it is simultaneously bounded below and above.
Now we are ready to define the concepts of infimum and supremum of sets.
Definition 1.15 Let D R be nonempty and bounded below. e infimum of D, denoted by inf D, is the greatest lower bound of D. When D is nonempty and bounded above, its supremum, denoted by sup D, is the least upper bound of D. If D is not bounded below .resp. above/, then we set inf D WD 1 .resp. sup D /. We also use the convention that inf and sup . WD 1 ; WD 1 ; WD 1 e following fundamental axiom ensures that these notions are well-defined. Completeness Axiom. For every nonempty subset D of R that is bounded above, the least upper bound of D exists as a real number. Using the Completeness Axiom, it is easy to see that if a nonempty set is bounded below, then its greatest lower bound exists as a real number. roughout the book we consider for convenience extended-real-valued functions, which take values in R . ; . e usual conventions of extended arithmetics are that a WD 1 1 C 1 D for any a R, , and t for t > 0. 1 2 1 C 1 D 1 1 D 1
Definition 1.16 Let f ˝ R be an extended-real-valued function and let x ˝ with f .x/ < W ! N 2 N . en f is at x if for any > 0 there is ı > 0 such that 1 N f .x/ f .x/ < whenever x x < ı; x ˝: j N j k Nk 2 We say that f is continuous on ˝ if it is continuous at every point of ˝.
It is obvious from the definition that if f ˝ R is continuous at x (with f .x/ < ), W ! N N 1 then it is finite on the intersection of ˝ and a ball centered at x with some radius r > 0. Further- N more, f ˝ R is continuous at x (with f .x/ < ) if and only if for every sequence xk in ˝ W ! N N 1 f g converging to x the sequence f .xk/ converges to f .x/. N f g N 1.2. CONVEX SETS 5 Definition 1.17 Let f ˝ R and let x ˝ with f .x/ < . We say that f has a - W ! N 2 N 1 at x relative to ˝ if there is ı > 0 such that N f .x/ f .x/ for all x IB.x ı/ ˝: N 2 NI \ We also say that f has a / at x relative to ˝ if N f .x/ f .x/ for all x ˝: N 2 e notions of local and global maxima can be defined similarly. Finally in this section, we formulate a fundamental result of mathematical analysis and optimization known as the Weierstrass existence theorem.
eorem 1.18 Let f ˝ R be a continuous function, where ˝ is a nonempty, compact subset W ! of Rn. en there exist x ˝ and u ˝ such that N 2 N 2 f .x/ inf ˚f .x/ ˇ x ˝« and f .u/ sup ˚f .x/ ˇ x ˝«: N D ˇ 2 N D ˇ 2 In Section 4.1 we present some “unilateral” versions of eorem 1.18.
1.2 CONVEX SETS We start the study of convexity with sets and then proceed to functions. Geometric ideas play an underlying role in convex analysis, its extensions, and applications. us we implement the geometric approach in this book. Given two elements a and b in Rn, define the interval/line segment
Œa; b ˚a .1 /b ˇ Œ0; 1«: WD C ˇ 2 Note that if a b, then this interval reduces to a singleton Œa; b a . D D f g
Definition 1.19 A subset ˝ of Rn is if Œa; b ˝ whenever a; b ˝. Equivalently, ˝ is 2 convex if a .1 /b ˝ for all a; b ˝ and Œ0; 1. C 2 2 2 n m m Given !1;:::;!m R , the element x Pi 1 i !i , where Pi 1 i 1 and i 0 for 2 D D D D some m N, is called a convex combination of !1;:::;!m. 2
Proposition 1.20 A subset ˝ of Rn is convex if and only if it contains all convex combinations of its elements. 6 1. CONVEX SETS AND FUNCTIONS
Figure 1.1: Convex set and nonconvex set.
Proof. e sufficient condition is trivial. To justify the necessity, we show by induction that any m convex combination x P i !i of elements in ˝ is an element of ˝. is conclusion follows D i 1 directly from the definition forD m 1; 2. Fix now a positive integer m 2 and suppose that every D convex combination of k N elements from ˝, where k m, belongs to ˝. Form the convex 2 Ä combination m 1 m 1 C C y X i !i ; X i 1; i 0 WD D i 1 i 1 D D and observe that if m 1 1, then 1 2 ::: m 0, so y !m 1 ˝. In the case C D D D D D D C 2 where m 1 < 1 we get the representations C m m i X i 1 m 1 and X 1; D C 1 m 1 D i 1 i 1 C D D which imply in turn the inclusion
m i z X !i ˝: WD 1 m 1 2 i 1 C D It yields therefore the relationships
m i y .1 m 1/ X !i m 1!m 1 .1 m 1/z m 1!m 1 ˝ D C 1 m 1 C C C D C C C C 2 i 1 C D and thus completes the proof of the proposition.
n p Proposition 1.21 Let ˝1 be a convex subset of R and let ˝2 be a convex subset of R . en the n p Cartesian product ˝1 ˝2 is a convex subset of R R . 1.2. CONVEX SETS 7
Proof. Fix a .a1; a2/; b .b1; b2/ ˝1 ˝2, and .0; 1/. en we have a1; b1 ˝1 and D D 2 2 2 a2; b2 ˝2. e convexity of ˝1 and ˝2 gives us 2
ai .1 /bi ˝i for i 1; 2; C 2 D which implies therefore that
a .1 /b a1 .1 /b1; a2 .1 /b2 ˝1 ˝2: C D C C 2
us the Cartesian product ˝1 ˝2 is convex. Let us continue with the definition of affine mappings.
Definition 1.22 A mapping B Rn Rp is if there exist a linear mapping A Rn Rp W ! W ! and an element b Rp such that B.x/ A.x/ b for all x Rn. 2 D C 2 Every linear mapping is affine. Moreover, B Rn Rp is affine if and only if W ! B.x .1 /y/ B.x/ .1 /B.y/ for all x; y Rn and R : C D C 2 2 Now we show that set convexity is preserved under affine operations.
Proposition 1.23 Let B Rn Rp be an affine mapping. Suppose that ˝ is a convex subset of Rn W p ! p 1 and is a convex subset of R . en B.˝/ is a convex subset of R and B ./ is a convex subset of Rn.
Proof. Fix any a; b B.˝/ and .0; 1/. en a B.x/ and b B.y/ for x; y ˝. Since ˝ 2 2 D D 2 is convex, we have x .1 /y ˝. en C 2 a .1 /b B.x/ .1 /B.y/ B.x .1 /y/ B.˝/; C D C D C 2 which justifies the convexity of the image B.˝/. Taking now any x; y B 1./ and .0; 1/, we get B.x/; B.y/ . is gives us 2 2 2 B.x/ .1 /B.y/ B x .1 /y C D C 2 1 by the convexity of . us we have x .1 /y B ./, which verifies the convexity of 1 C 2 the inverse image B ./.
n Proposition 1.24 Let ˝1; ˝2 R be convex and let R. en both sets ˝1 ˝2 and ˝1 2 C are also convex in Rn.
Proof. It follows directly from the definitions. 8 1. CONVEX SETS AND FUNCTIONS Next we proceed with intersections of convex sets.
n Proposition 1.25 Let ˝˛ ˛ I be a collection of convex subsets of R . en T˛ I ˝˛ is also a convex f g 2 subset of Rn. 2
Proof. Taking any a; b T˛ I ˝˛, we get that a; b ˝˛ for all ˛ I . e convexity of each 2 2 2 2 ˝˛ ensures that a .1 /b ˝˛ for any .0; 1/. us a .1 /b T˛ I ˝˛ and C 2 2 C 2 2 the intersection T˛ I ˝˛ is convex. 2
Definition 1.26 Let ˝ be a subset of Rn. e of ˝ is defined by
n ˇ o co ˝ \ C ˇ C is convex and ˝ C : WD ˇ e next proposition follows directly from the definition and Proposition 1.25.
Figure 1.2: Nonconvex set and its convex hull.
Proposition 1.27 e convex hull co ˝ is the smallest convex set containing ˝.
Proof. e convexity of the set co ˝ ˝ follows from Proposition 1.25. On the other hand, for any convex set C that contains ˝ we clearly have co ˝ C , which verifies the conclusion.
Proposition 1.28 For any subset ˝ of Rn, its convex hull admits the representation
m m
co ˝ n X i ai ˇ X i 1; i 0; ai ˝; m No: D ˇ D 2 2 i 1 ˇ i 1 D D 1.2. CONVEX SETS 9 Proof. Denoting by C the right-hand side of the representation to prove, we obviously have ˝ C . Let us check that the set C is convex. Take any a; b C and get 2 p q
a X ˛i ai ; b X ˇj bj ; WD WD i 1 j 1 D D p q where ai ; bj ˝, ˛i ; ˇj 0 with P ˛i P ˇj 1, and p; q N. It follows easily that 2 i 1 D j 1 D 2 for every number .0; 1/, we have D D 2 p q
a .1 /b X ˛i ai X.1 /ˇj bj : C D C i 1 j 1 D D en the resulting equality
p q p q
X ˛i X.1 /ˇj X ˛i .1 / X ˇj 1 C D C D i 1 j 1 i 1 j 1 D D D D ensures that a .1 /b C , and thus co ˝ C by the definition of co ˝. Fix now any a m C 2 D P i ai C with ai ˝ co ˝ for i 1; : : : ; m. Since the set co ˝ is convex, we conclude i 1 2 2 D by PropositionD 1.20 that a co ˝ and thus co ˝ C . 2 D
Proposition 1.29 e interior int ˝ and closure ˝ of a convex set ˝ Rn are also convex.
Proof. Fix a; b int ˝ and .0; 1/. en find an open set V such that 2 2 a V ˝ and so a .1 /b V .1 /b ˝: 2 C 2 C Since V .1 /b is open, we get a .1 /b int ˝, and thus the set int ˝ is convex. C C 2 To verify the convexity of ˝, we fix a; b ˝ and .0; 1/ and then find sequences ak 2 2 f g and bk in ˝ converging to a and b, respectively. By the convexity of ˝, the sequence ak f g f C .1 /bk lies entirely in ˝ and converges to a .1 /b. is ensures the inclusion a g C C .1 /b ˝ and thus justifies the convexity of the closure ˝. 2 To proceed further, for any a; b Rn, define the interval 2 Œa; b/ ˚a .1 /b ˇ .0; 1«: WD C ˇ 2 We can also define the intervals .a; b and .a; b/ in a similar way.
Lemma 1.30 For a convex set ˝ Rn with nonempty interior, take any a int ˝ and b ˝. 2 2 en Œa; b/ int ˝. 10 1. CONVEX SETS AND FUNCTIONS
Proof. Since b ˝, for any > 0, we have b ˝ IB. Take now .0; 1 and let x 2 22 C 2 WD a .1 /b. Choosing > 0 such that a IB ˝ gives us C C
x IB a .1 /b IB C D C C a .1 /˝ IB IB C C C a .1 /˝ .1 /IB IB D C 2 C C ha IBi .1 /˝ C C ˝ .1 /˝ ˝: C
is shows that x int ˝ and thus verifies the inclusion Œa; b/ int ˝. 2 Now we establish relationships between taking the interior and closure of convex sets.
Proposition 1.31 Let ˝ Rn be a convex set with nonempty interior. en we have: (i) int ˝ ˝ and .ii/ int ˝ int ˝. D D
Proof. (i) Obviously, int ˝ ˝. For any b ˝ and a int ˝, define the sequence xk by 2 2 f g 1 1 xk a 1 Áb; k N: WD k C k 2
Lemma 1.30 ensures that xk int ˝. Since xk b, we have b int ˝ and thus verify (i). 2 ! 2 (ii) Since the inclusion int ˝ int ˝ is obvious, it remains to prove the opposite inclusion int ˝ int ˝. To proceed, fix any b int ˝ and a int ˝. If > 0 is sufficiently small, then 2 2 1 c b .b a/ ˝, and hence b a c .a; c/ int ˝; WD C 2 D 1 C 1 2 C C which verifies that int ˝ int ˝ and thus completes the proof. 1.3 CONVEX FUNCTIONS is section collects basic facts about general (extended-real-valued) convex functions including their analytic and geometric characterizations, important properties as well as their specifications for particular subclasses. We also define convex set-valued mappings and use them to study a re- markable class of optimal value functions employed in what follows.
Definition 1.32 Let f ˝ R be an extended-real-valued function define on a convex set ˝ W ! Rn. en the function f is on ˝ if
f x .1 /y f.x/ .1 /f .y/ for all x; y ˝ and .0; 1/: (1.1) C Ä C 2 2 1.3. CONVEX FUNCTIONS 11 If the inequality in (1.1) is strict for x y, then f is on ˝. ¤ Given a function f ˝ R, the extension of f to Rn is defined by W ! (f .x/ if x ˝; f .x/ 2 e WD otherwise: 1 Obviously, if f is convex on ˝, where ˝ is a convex set, then f is convex on Rn. Furthermore, e if f Rn R is a convex function, then it is also convex on every convex subset of Rn. is W ! allows to consider without loss of generality extended-real-valued convex functions on the whole space Rn.
Figure 1.3: Convex function and nonconvex function.
Definition 1.33 e and of f Rn R are defined, respectively, by W ! dom f ˚x Rn ˇ f .x/ < « and WD 2 ˇ 1 epi f ˚.x; t/ Rn R ˇ x Rn; t f .x/« ˚.x; t/ Rn R ˇ x dom f; t f .x/«: WD 2 ˇ 2 D 2 ˇ 2 Let us illustrate the convexity of functions by examples.
Example 1.34 e following functions are convex: (i) f .x/ a; x b for x Rn, where a Rn and b R. WD h i C 2 2 2 (ii) g.x/ x for x Rn. WD k k 2 (iii) h.x/ x2 for x R. WD 2 12 1. CONVEX SETS AND FUNCTIONS Indeed, the function f in (i) is convex since
f x .1 /y a; x .1 /y b a; x .1 / a; y b C D h C i C D h i C h i C . a; x b/ .1 /. a; y b/ D h i C C h i C f.x/ .1 /f .y/ for all x; y Rn and .0; 1/: D C 2 2 e function g in (ii) is convex since for x; y Rn and .0; 1/, we have 2 2 g x .1 /y x .1 /y x .1 / y g.x/ .1 /g.y/; C D k C k Ä k k C k k D C which follows from the triangle inequality and the fact that ˛u ˛ u whenever ˛ R k k D j j k k 2 and u Rn. e convexity of the simplest quadratic function h in (iii) follows from a more general 2 result for the quadratic function on Rn given in the next example.
Example 1.35 Let A be an n n symmetric matrix. It is called positive semidefinite if Au; u 0 h i for all u Rn. Let us check that A is positive semidefinite if and only if the function f Rn R 2 W ! defined by 1 f .x/ Ax; x ; x Rn; WD 2h i 2 is convex. Indeed, a direct calculation shows that for any x; y Rn and .0; 1/ we have 2 2 1 f.x/ .1 /f .y/ f x .1 /y .1 / A.x y/; x y : (1.2) C C D 2 h i If the matrix A is positive semidefinite, then A.x y/; x y 0, so the function f is convex h i by (1.2). Conversely, assuming the convexity of f and using equality (1.2) for x u and y 0 D D verify that A is positive semidefinite.
e following characterization of convexity is known as the Jensen inequality.
n eorem 1.36 A function f R R is convex if and only if for any numbers i 0 as i m W ! n D 1; : : : ; m with Pi 1 i 1 and for any elements xi R , i 1; : : : ; m, it holds that D D 2 D m m
f X i xi Á X i f .xi /: (1.3) Ä i 1 i 1 D D Proof. Since (1.3) immediately implies the convexity of f , we only need to prove that any convex function f satisfies the Jensen inequality (1.3). Arguing by induction and taking into account that for m 1 inequality (1.3) holds trivially and for m 2 inequality (1.3) holds by the definition D D of convexity, we suppose that it holds for an integer m k with k 2. Fix numbers i 0, k 1 WDn i 1; : : : ; k 1, with P C i 1 and elements xi R , i 1; : : : ; k 1. We obviously have D C i 1 D 2 D C the relationship D k
X i 1 k 1: D C i 1 D 1.3. CONVEX FUNCTIONS 13
If k 1 1, then i 0 for all i 1; : : : ; k and (1.3) obviously holds for m k 1 in this C D D D WD C case. Supposing now that 0 k 1 < 1, we get Ä C k X i 1 1 D i 1 k 1 D C and by direct calculations based on convexity arrive at
k 1 k C Pi 1 i xi f X i xi Á f h.1 k 1/ D k 1xk 1i D C 1 k 1 C C C i 1 C D k Pi 1 i xi .1 k 1/f D Á k 1f .xk 1/ Ä C 1 k 1 C C C k C i .1 k 1/f X xi Á k 1f .xk 1/ D C 1 k 1 C C C i 1 C k D i .1 k 1/ X f .xi / k 1f .xk 1/ Ä C 1 k 1 C C C i 1 C k 1 D C X i f .xi /: D i 1 D is justifies inequality (1.3) and completes the proof of the theorem. e next theorem gives a geometric characterization of the function convexity via the con- vexity of the associated epigraphical set.
eorem 1.37 A function f Rn R is convex if and only if its epigraph epi f is a convex W ! subset of the product space Rn R.
Proof. Assuming that f is convex, fix pairs .x1; t1/; .x2; t2/ epi f and a number .0; 1/. 2 2 en we have f .xi / ti for i 1; 2. us the convexity of f ensures that Ä D f x1 .1 /x2 f.x1/ .1 /f .x2/ t1 .1 /t2: C Ä C Ä C is implies therefore that
.x1; t1/ .1 /.x2; t2/ .x1 .1 /x2; t1 .1 /t2/ epi f; C D C C 2 and thus the epigraph epi f is a convex subset of Rn R. Conversely, suppose that the set epi f is convex and fix x1; x2 dom f and a number 2 .0; 1/. en .x1; f .x1//; .x2; f .x2// epi f . is tells us that 2 2 x1 .1 /x2; f.x1/ .1 /f .x2/ x1; f .x1/ .1 / x2; f .x2/ epi f C C D C 2 14 1. CONVEX SETS AND FUNCTIONS and thus implies the inequality
f x1 .1 /x2 f.x1/ .1 /f .x2/; C Ä C which justifies the convexity of the function f .
Figure 1.4: Epigraphs of convex function and nonconvex function.
Now we show that convexity is preserved under some important operations.
n Proposition 1.38 Let fi R R be convex functions for all i 1; : : : ; m. en the following W ! D functions are convex as well: (i) e multiplication by scalars f for any > 0. m (ii) e sum function Pi 1 fi . D (iii) e maximum function max1 i m fi . Ä Ä
Proof. e convexity of f in (i) follows directly from the definition. It is sufficient to prove (ii) and (iii) for m 2, since the general cases immediately follow by induction. D (ii) Fix any x; y Rn and .0; 1/. en we have 2 2 f1 f2 x .1 /y f1 x .1 /y f2 x .1 /y C C D C C C f1.x/ .1 /f1.y/ f2.x/ .1 /f2.y/ Ä C C C .f1 f2/.x/ .1 /.f1 f2/.y/; D C C C which verifies the convexity of the sum function f1 f2. C n (iii) Denote g max f1; f2 and get for any x; y R and .0; 1/ that WD f g 2 2 fi x .1 /y fi .x/ .1 /fi .y/ g.x/ .1 /g.y/ C Ä C Ä C for i 1; 2. is directly implies that D g x .1 /y max ˚f1 x .1 /y; f2 x .1 /y« g.x/ .1 /g.y/; C D C C Ä C which shows that the maximum function g.x/ max f1.x/; f2.x/ is convex. D f g 1.3. CONVEX FUNCTIONS 15 e next result concerns the preservation of convexity under function compositions.
Proposition 1.39 Let f Rn R be convex and let ' R R be nondecreasing and convex on W ! W ! a convex set containing the range of the function f . en the composition ' f is convex. ı
n Proof. Take any x1; x2 R and .0; 1/. en we have by the convexity of f that 2 2 f x1 .1 /x2 f.x1/ .1 /f .x2/: C Ä C Since ' is nondecreasing and it is also convex, it follows that
' f x1 .1 /x2 ' f x1 .1 /x2 ı C D C ' f.x1/ .1 /f .x2/ Ä C ' f .x1/ .1 /' f .x2/ Ä C .' f /.x1/ .1 /.' f /.x2/; D ı C ı which verifies the convexity of the composition ' f . ı Now we consider the composition of a convex function and an affine mapping.
Proposition 1.40 Let B Rn Rp be an affine mapping and let f Rp R be a convex func- W ! W ! tion. en the composition f B is convex. ı
Proof. Taking any x; y Rn and .0; 1/, we have 2 2 f B x .1 /y f .B.x .1 /y// f B.x/ .1 /B.y/ ı C D C D C f B.x/ .1 /f B.y/ .f B/.x/ .1 /.f B/.y/ Ä C D ı C ı and thus justify the convexity of the composition f B. ı e following simple consequence of Proposition 1.40 is useful in applications.
Corollary 1.41 Let f Rn R be a convex function. For any x; d Rn, the function ' R R W ! N 2 W ! defined by '.t/ f .x td/ is convex as well. Conversely, if for every x; d Rn the function ' WD N C N 2 defined above is convex, then f is also convex.
Proof. Since B.t/ x td is an affine mapping, the convexity of ' immediately follows from DN C n Proposition 1.40. To prove the converse implication, take any x1; x2 R , .0; 1/ and let 2 2 x x2, d x1 x2. Since '.t/ f .x td/ is convex, we have N WD WD D N C f x1 .1 /x2 f x2 .x1 x2/ './ ' .1/ .1 /.0/ C D C D D C '.1/ .1 /'.0/ f.x1/ .1 /f .x2/; Ä C D C which verifies the convexity of the function f . 16 1. CONVEX SETS AND FUNCTIONS e next proposition is trivial while useful in what follows.
Proposition 1.42 Let f Rn Rp R be convex. For .x; y/ Rn Rp, the functions '.y/ W ! N N 2 WD f .x; y/ and .x/ f .x; y/ are also convex. N WD N Now we present an important extension of Proposition 1.38(iii).
n Proposition 1.43 Let fi R R for i I be a collection of convex functions with a nonempty W ! 2 index set I . en the supremum function f .x/ supi I fi .x/ is convex. WD 2
n Proof. Fix x1; x2 R and .0; 1/. For every i I , we have 2 2 2
fi x1 .1 /x2 fi .x1/ .1 /fi .x2/ f.x1/ .1 /f .x2/; C Ä C Ä C which implies in turn that
f x1 .1 /x2 sup fi x1 .1 /x2 f.x1/ .1 /f .x2/: C D i I C Ä C 2 is justifies the convexity of the supremum function. Our next intention is to characterize convexity of smooth functions of one variable. To pro- ceed, we begin with the following lemma.
Lemma 1.44 Given a convex function f R R, assume that its domain is an open interval I . W ! For any a; b I and a < x < b, we have the inequalities 2 f .x/ f .a/ f .b/ f .a/ f .b/ f .x/ : x a Ä b a Ä b x x a Proof. Fix a; b; x as above and form the numbers t .0; 1/. en WD b a 2 x a f .x/ f a .x a/ f a b aÁ f a t.b a/ f tb .1 t/a: D C D C b a D C D C is gives us the inequalities f .x/ tf .b/ .1 t/f .a/ and Ä C x a f .x/ f .a/ tf .b/ .1 t/f .a/ f .a/ tf .b/ f .a/ f .b/ f .a/Á; Ä C D D b a which can be equivalently written as
f .x/ f .a/ f .b/ f .a/ : x a Ä b a 1.3. CONVEX FUNCTIONS 17 Similarly, we have the estimate x b f .x/ f .b/ tf .b/ .1 t/f .a/ f .b/ .t 1/f .b/ f .a/ f .b/ f .a/Á; Ä C D D b a which finally implies that f .b/ f .a/ f .b/ f .x/ b a Ä b x and thus completes the proof of the lemma.
eorem 1.45 Suppose that f R R is differentiable on its domain, which is an open interval W ! I . en f is convex if and only if the derivative f 0 is nondecreasing on I .
Proof. Suppose that f is convex and fix a < b with a; b I . By Lemma 1.44, we have 2 f .x/ f .a/ f .b/ f .a/ x a Ä b a for any x .a; b/. is implies by the derivative definition that 2 f .b/ f .a/ f 0.a/ : Ä b a Similarly, we arrive at the estimate f .b/ f .a/ f 0.b/ b a Ä and conclude that f .a/ f .b/, i.e., f is a nondecreasing function. 0 Ä 0 0 To prove the converse, suppose that f is nondecreasing on I and fix x1 < x2 with x1; x2 0 2 I and t .0; 1/. en 2 x1 < xt < x2 for xt tx1 .1 t/x2: WD C By the mean value theorem, we find c1; c2 such that x1 < c1 < xt < c2 < x2 and
f .xt / f .x1/ f 0.c1/.xt x1/ f 0.c1/.1 t/.x2 x1/; D D f .xt / f .x2/ f 0.c2/.xt x2/ f 0.c2/t.x1 x2/: D D is can be equivalently rewritten as
tf .xt / tf .x1/ f 0.c1/t.1 t/.x2 x1/; D .1 t/f .xt / .1 t/f .x2/ f 0.c2/t.1 t/.x1 x2/: D Since f .c1/ f .c2/, adding these equalities yields 0 Ä 0 f .xt / tf .x1/ .1 t/f .x2/; Ä C which justifies the convexity of the function f . 18 1. CONVEX SETS AND FUNCTIONS Corollary 1.46 Let f R R be twice differentiable on its domain, which is an open interval I . W ! en f is convex if and only if f .x/ 0 for all x I . 00 2
Proof. Since f .x/ 0 for all x I if and only if the derivative function f is nondecreasing on 00 2 0 this interval. en the conclusion follows directly from eorem 1.45.
Example 1.47 Consider the function
1 8 if x > 0; f .x/
Definition 1.48 We say that F is a - between Rn and Rp and denote it by F Rn Rp if F .x/ is a subset of Rp for every x Rn. e and of F are defined, W ! 2 respectively,! by
dom F ˚x Rn ˇ F .x/ « and gph F ˚.x; y/ Rn Rp ˇ y F.x/«: WD 2 ˇ ¤ ; WD 2 ˇ 2
Any single-valued mapping F Rn Rp is a particular set-valued mapping where the set W ! F.x/ is a singleton for every x Rn. It is essential in the following definition that the mapping 2 F is set-valued.
Definition 1.49 Let F Rn Rp and let ' Rn Rp R. e or W ! W ! associated with F and! ' is defined by
.x/ inf ˚'.x; y/ ˇ y F .x/«; x Rn : (1.4) WD ˇ 2 2 roughout this section we assume that .x/ > for every x Rn. 1 2
Proposition 1.50 Assume that ' Rn Rp R is a convex function and that F Rn Rp is of W ! W ! convex graph. en the optimal value function in (1.4) is convex. ! 1.3. CONVEX FUNCTIONS 19
Figure 1.5: Graph of set-valued mapping.
Proof. Take x1; x2 dom , .0; 1/. For any > 0, find yi F.xi / such that 2 2 2 '.xi ; yi / < .xi / for i 1; 2: C D It directly implies the inequalities
'.x1; y1/ < .x1/ ; .1 /'.x2; y2/ < .1 /.x2/ .1 /: C C Summing up these inequalities and employing the convexity of ' yield
' x1 .1 /x2; y1 .1 /y2 '.x1; y1/ .1 /'.x2; y2/ C C Ä C < .x1/ .1 /.x2/ : C C Furthermore, the convexity of gph F gives us
x1 .1 /x2; y1 .1 /y2 .x1; y1/ .1 /.x2; y2/ gph F; C C D C 2 and therefore y1 .1 /y2 F .x1 .1 /x2/. is implies that C 2 C x1 .1 /x2 ' x1 .1 /x2; y1 .1 /y2 < .x1/ .1 /.x2/ : C Ä C C C C Letting finally 0 ensures the convexity of the optimal value function . ! Using Proposition 1.50, we can verify convexity in many situations. For instance, given n two convex functions fi R R, i 1; 2, let '.x; y/ f1.x/ y and F .x/ Œf2.x/; /. W ! D WD C WD 1 en the function ' is convex and set gph F epi f2 is convex as well, and hence we justify the D convexity of the sum .x/ inf '.x; y/ f1.x/ f2.x/: D y F .x/ D C 2 20 1. CONVEX SETS AND FUNCTIONS Another example concerns compositions. Let f Rp R be convex and let B Rn Rp be W ! W ! affine. Define '.x; y/ f .y/ and F.x/ B.x/ . Observe that ' is convex while F is of con- WD WD f g vex graph. us we have the convex composition
.x/ inf '.x; y/ f B.x/; x Rn : D y F .x/ D 2 2 e examples presented above recover the results obtained previously by direct proofs. Now we establish via Proposition 1.50 the convexity of three new classes of functions.
Proposition 1.51 Let ' Rp R be convex and let B Rp Rn be affine. Consider the set- W !1 n p W ! valued inverse image mapping B R R , define W ! 1 n f .x/ inf ˚'.y/ ˇ y B .x/«; x R ; WD ˇ 2 2 and suppose that f .x/ > for all x Rn. en f is a convex function. 1 2
Proof. Let '.x; y/ '.y/ and F.x/ B 1.x/. en the set Á WD gph F ˚.u; v/ Rn Rp ˇ B.v/ u« D 2 ˇ D is obviously convex. Since we have the representation
f .x/ inf '.y/; x Rn; D y F .x/ 2 2 the convexity of f follows directly from Proposition 1.50.
n Proposition 1.52 For convex functions f1; f2 R R, define the W !
.f1 f2/.x/ inf ˚f1.x1/ f2.x2/ ˇ x1 x2 x« ˚ WD C ˇ C D n and suppose that .f1 f2/.x/ > for all x R . en f1 f2 is also convex. ˚ 1 2 ˚
n n n n n Proof. Define ' R R R by '.x1; x2/ f1.x1/ f2.x2/ and B R R R by W ! WD C W ! B.x1; x2/ x1 x2. We have WD C 1 n inf ˚'.x1; x2/ ˇ .x1; x2/ B .x/« .f1 f2/.x/ for all x R ; ˇ 2 D ˚ 2
which implies the convexity of .f1 f2/ by Proposition 1.51. ˚ 1.4. RELATIVE INTERIORS OF CONVEX SETS 21 Definition 1.53 A function g Rp R is called if W !
xi yi for all i 1; : : : ; p g.x1; : : : ; xp/ g.y1; : : : ; yp/: Ä D H) Ä Now we are ready to present the final consequence of Proposition 1.50 in this section that involves the composition.
n p n Proposition 1.54 Define h R R by h.x/ .f1.x/; : : : ; fp.x//, where fi R R for W ! WD W ! i 1; : : : ; p are convex functions. Suppose that g Rp R is convex and nondecreasing componen- D W ! twise. en the composition g h Rn R is a convex function. ı W !
Proof. Let F Rn Rp be a set-valued mapping defined by W ! F .x/ Œf1.x/; / Œf2.x/; / ::: Œfp.x/; /: WD 1 1 1 en the graph of F is represented by
n p gph F ˚.x; t1; : : : ; tp/ R R ˇ ti fi .x/«: D 2 ˇ n p Since all fi are convex, the set gph F is convex as well. Define further ' R R R by W ! '.x; y/ g.y/ and observe, since g is increasing componentwise, that WD
inf ˚'.x; y/ ˇ y F.x/« g f1.x/; : : : ; fp.x/ .g h/.x/; ˇ 2 D D ı which ensures the convexity of the composition g h by Proposition 1.50. ı 1.4 RELATIVE INTERIORS OF CONVEX SETS We begin this section with the definition and properties of affine sets. Given two elements a and b in Rn, the line connecting them is
LŒa; b ˚a .1 /b ˇ R «: WD C ˇ 2 Note that if a b, then LŒa; b a . D D f g Definition 1.55 A subset ˝ of Rn is if for any a; b ˝ we have LŒa; b ˝. 2 For instance, any point, line, and plane in R3 are affine sets. e empty set and the whole space are always affine. It follows from the definition that the intersection of any collection of affine sets is affine. is leads us to the construction of the affine hull of a set.
Definition 1.56 e of a set ˝ Rn is aff ˝ \ ˚C ˇ C is affine and ˝ C «: WD ˇ 22 1. CONVEX SETS AND FUNCTIONS An element x in Rn of the form
m m
x X i !i with X i 1; m N; D D 2 i 1 i 1 D D
is called an affine combination of !1;:::;!m. e proof of the next proposition is straightforward and thus is omitted.
Proposition 1.57 e following assertions hold: (i) A set ˝ is affine if and only if ˝ contains all affine combinations of its elements. n (ii) Let ˝, ˝1, and ˝2 be affine subsets of R . en the sum ˝1 ˝2 and the scalar product ˝ for C any R are also affine subsets of Rn. 2 (iii) Let B Rn Rp be an affine mapping. If ˝ is an affine subset of Rn and is an affine subset p W ! p 1 of R , then the image B.˝/ is an affine subset of R and the inverse image B ./ is an affine subset of Rn. (iv) Given ˝ Rn, its affine hull is the smallest affine set containing ˝. We have m m
aff ˝ n X i !i ˇ X i 1; !i ˝; m N o: D ˇ D 2 2 i 1 ˇ i 1 D D (v) A set ˝ is a .linear/ subspace if and only if ˝ is an affine set containing the origin.
Next we consider relationships between affine sets and (linear) subspaces.
Lemma 1.58 A nonempty subset ˝ of Rn is affine if and only if ˝ ! is a subspace of Rn for any ! ˝. 2 Proof. Suppose that a nonempty set ˝ Rn is affine. It follows from Proposition 1.57(v) that ˝ ! is a subspace for any ! ˝. Conversely, fix ! ˝ and suppose that ˝ ! is a subspace 2 2 denoted by L. en the set ˝ ! L is obviously affine. D C e preceding lemma leads to the following notion.
Definition 1.59 An affine set ˝ is to a subspace L if ˝ ! L for some ! ˝. D C 2 e next proposition justifies the form of the parallel subspace.
Proposition 1.60 Let ˝ be a nonempty, affine subset of Rn. en it is parallel to the unique subspace L of Rn given by L ˝ ˝. D 1.4. RELATIVE INTERIORS OF CONVEX SETS 23 Proof. Given a nonempty, affine set ˝, fix ! ˝ and come up to the linear subspace L ˝ ! 2 WD parallel to ˝. To justify the uniqueness of such L, take any !1;!2 ˝ and any subspaces n 2 L1;L2 R such that ˝ !1 L1 !2 L2. en L1 !2 !1 L2. Since 0 L1, we D C D C D C 2 have !1 !2 L2. is implies that !2 !1 L2 and thus L1 !2 !1 L2 L2. Simi- 2 2 D C larly, we get L2 L1, which justifies that L1 L2. D It remains to verify the representation L ˝ ˝. Let ˝ ! L with the unique sub- D D C space L and some ! ˝. en L ˝ ! ˝ ˝. Fix any x u ! with u; ! ˝ and 2 D D 2 observe that ˝ ! is a subspace parallel to ˝. Hence ˝ ! L by the uniqueness of L proved D above. is ensures that x ˝ ! L and thus ˝ ˝ L. 2 D e uniqueness of the parallel subspace shows that the next notion is well defined.
Definition 1.61 e ˝ Rn is the dimension of the linear ; ¤ subspace parallel to ˝. Furthermore, the ˝ Rn is the dimension ; ¤ of its affine hull aff ˝.
To proceed further, we need yet another definition important in what follows.
n Definition 1.62 e elements v0; : : : ; vm in R , m 1, are if m m h X i vi 0; X i 0i i 0 for all i 0; : : : ; m: D D H) D D i 0 i 0 D D
It is easy to observe the following relationship with the linear independence.
n Proposition 1.63 e elements v0; : : : ; vm in R are affinely independent if and only if the elements v1 v0; : : : ; vm v0 are linearly independent.
Proof. Suppose that v0; : : : ; vm are affinely independent and consider the system
m m
X i .vi v0/ 0; i.e., 0v0 X i vi 0; D C D i 1 i 1 D D m m where 0 Pi 1 i . Since the elements v0; : : : ; vm are affinely independent and Pi 0 i WD D D D 0, we have that i 0 for all i 1; : : : ; m. us v1 v0; : : : ; vm v0 are linearly independent. D D e proof of the converse statement is straightforward. Recall that the span of some set C , span C , is the linear subspace generated by C .
n Lemma 1.64 Let ˝ aff v0; : : : ; vm , where vi R for all i 0; : : : ; m. en the span of the WD f g 2 D set v1 v0; : : : ; vm v0 is the subspace parallel to ˝. f g 24 1. CONVEX SETS AND FUNCTIONS
Proof. Denote by L the subspace parallel to ˝. en ˝ v0 L and therefore vi v0 L for D 2 all i 1; : : : ; m. is gives D
span ˚vi v0 ˇ i 1; : : : ; m« L: ˇ D
To prove the converse inclusion, fix any v L and get v v0 ˝. us we have 2 C 2 m m
v v0 X i vi ; X i 1: C D D i 0 i 0 D D is implies the relationship
m v X i .vi v0/ span ˚vi v0 ˇ i 1; : : : ; m«; D 2 ˇ D i 1 D which justifies the converse inclusion and hence completes the proof. e proof of the next proposition is rather straightforward.
n Proposition 1.65 e elements v0; : : : ; vm are affinely independent in R if and only if its affine hull ˝ aff v0; : : : ; vm is m-dimensional. WD f g
Proof. Suppose that v0; : : : ; vm are affinely independent. en Lemma 1.64 tells us that the subspace L span vi v0 i 1; : : : ; m is parallel to ˝. e linear independence of v1 WD f j D g v0; : : : ; vm v0 by Proposition 1.63 means that the subspace L is m-dimensional and so is ˝. e proof of the converse statement is also straightforward. Affinely independent systems lead us to the construction of simplices.
n Definition 1.66 Let v0; : : : ; vm be affinely independent in R . en the set
m co˚vi ˇ i 0; : : : ; m« WD ˇ D n is called an m- in R with the vertices vi , i 0; : : : ; m. D An important role of simplex vertices is revealed by the following proposition.
Proposition 1.67 Consider an m-simplex m with vertices vi for i 0; : : : ; m. For every v m, m 1 D 2 there is a unique element .0; : : : ; m/ R C such that 2 C m m
v X i vi ; X i 1: D D i 0 i 0 D D 1.4. RELATIVE INTERIORS OF CONVEX SETS 25 m 1 m 1 Proof. Let .0; : : : ; m/ R C and .0; : : : ; m/ R C satisfy 2 C 2 C m m m m
v X i vi X i vi ; X i X i 1: D D D D i 0 i 0 i 0 i 0 D D D D is immediately implies the equalities
m m
X.i i /vi 0; X.i i / 0: D D i 0 i 0 D D
Since v0; : : : ; vm are affinely independent, we have i i for i 0; : : : ; m. D D Now we are ready to define a major notion of relative interiors of convex sets.
Definition 1.68 Let ˝ be a convex set. We say that an element v ˝ belongs to the 2 ri ˝ of ˝ if there exists > 0 such that IB.v / aff ˝ ˝. I \ We begin the study of relative interiors with the following lemma.
Lemma 1.69 Any linear mapping A Rn Rp is continuous. W !
n Proof. Let e1; : : : ; en be the standard orthonormal basis of R and let vi A.ei /, i 1; : : : ; n. f n g WD D For any x R with x .x1; : : : ; xn/, we have 2 D n n n
A.x/ A X xi ei Á X xi A.ei / X xi vi : D D D i 1 i 1 i 1 D D D en the triangle inequality and the Cauchy-Schwarz inequality give us
n v n v n u 2u 2 A.x/ X xi vi uX xi uX vi M x ; k k Ä j jk k Ä t j j t k k D k k i 1 i 1 i 1 D D D
q n 2 where M Pi 1 vi . It follows furthermore that WD D k k A.x/ A.y/ A.x y/ M x y for all x; y Rn; k k D k k Ä k k 2 which implies the continuity of the mapping A. e next proposition plays an essential role in what follows.
n Proposition 1.70 Let m be an m-simplex in R with some m 1. en ri m . ¤ ; 26 1. CONVEX SETS AND FUNCTIONS
Proof. Consider the vertices v0; : : : ; vm of the simplex m and denote
1 m v X vi : WD m 1 i 0 C D
We prove the proposition by showing that v ri m. Define 2
L span˚vi v0 ˇ i 1; : : : ; m« WD ˇ D n and observe that L is the m-dimensional subspace of R parallel to aff m aff v0; : : : ; vm . It D mf 1 g is easy to see that for every x L there is a unique collection .0; : : : ; m/ R C with 2 2 m m
x X i vi ; X i 0: D D i 0 i 0 D D m 1 Consider the mapping A L R C , which maps x to the corresponding coefficients m 1 W ! .0; : : : ; m/ R C as above. en A is linear, and hence it is continuous by Lemma 1.69. 2 Since A.0/ 0, we can choose ı > 0 such that D 1 A.u/ < whenever u ı: k k m 1 k k Ä C Let us now show that .v ıIB/ aff m m, which means that v ri m. To proceed, fix any C \ 2 x .v ıIB/ aff m and get that x v u for some u ıIB. Since v; x aff m and u 2 C \ D C 2 2 m D x v, we have u L. Denoting A.u/ .˛0; : : : ; ˛m/ gives us the representation u Pi 0 ˛i vi m 2 WD D D with Pi 0 ˛i 0 and the estimate D D 1 ˛i A.u/ < for all i 0; : : : ; m: j j Ä k k m 1 D C en implies in turn that
m 1 m v u X. ˛i /vi X i vi ; C D m 1 C D i 0 i 0 D C D
1 m where i ˛i 0 for i 0; : : : ; m. Since Pi 0 i 1, this ensures that x m. WD m 1 C D D D 2 us .v ıIB/ Caff m m and therefore v ri m. C \ 2
Lemma 1.71 Let ˝ be a nonempty, convex set in Rn of dimension m 1. en there exist m 1 C affinely independent elements v0; : : : ; vm in ˝. 1.4. RELATIVE INTERIORS OF CONVEX SETS 27
Proof. Let k v0; : : : ; vk be a k simplex of maximal dimension contained in ˝. en WD f g v0; : : : ; vk are affinely independent. To verify now that k m, form K aff v0; : : : ; vk and D WD f g observe that K aff ˝ since v0; : : : ; vk ˝. e opposite inclusion also holds since we have f g ˝ K. Justifying it, we argue by contradiction and suppose that there exists w ˝ such that 2 w K. en a direct application of the definition of affine independence shows that v0; : : : ; vk; w … are affinely independent being a subset of ˝, which is a contradiction. us K aff ˝, and hence D we get k dim K dim aff ˝ dim ˝ m. D D D D e next is one of the most fundamental results of convex finite-dimensional geometry.
eorem 1.72 Let ˝ be a nonempty, convex set in Rn. e following assertions hold: (i) We always have ri ˝ . ¤ ; (ii) We have Œa; b/ ri ˝ for any a ri ˝ and b ˝. 2 2
Proof. (i) Let m be the dimension of ˝. Observe first that the case where m 0 is trivial since in D this case ˝ is a singleton and ri ˝ ˝. Suppose that m 1 and find m 1 affinely independent D C elements v0; : : : ; vm in ˝ as in Lemma 1.71. Consider further the m-simplex
m co˚v0; : : : ; vm«: WD
We can show that aff m aff ˝. To complete the proof, take v ri m, which exists by Propo- D 2 sition 1.70, and get for any small > 0 that
IB.v; / aff ˝ IB.v; / aff m m ˝: \ D \ is verifies that v ri ˝ by the definition of relative interior. 2 (ii) Let L be the subspace of Rn parallel to aff ˝ and let m dim L. en there is a bijective m 1 WD linear mapping A L R such that both A and A are continuous. Fix x0 aff ˝ and define W ! m 2 the mapping f aff ˝ R by f .x/ A.x x0/. It is easy to check that f is a bijective W ! WD affine mapping and that both f and f 1 are continuous. We also see that a ri ˝ if and only if 2 f .a/ int f .˝/, and that b ˝ if and only if f .b/ f .˝/. en Œf .a/; f .b// int f .˝/ by 2 2 2 Lemma 1.30. is shows that Œa; b/ ri ˝. We conclude this section by the following properties of the relative interior.
Proposition 1.73 Let ˝ be a nonempty, convex subset of Rn. For the convex sets ri ˝ and ˝, we have that (i) ri ˝ ˝ and (ii) ri ˝ ri ˝. D D 28 1. CONVEX SETS AND FUNCTIONS Proof. Note that the convexity of ri ˝ follows from eorem 1.72(ii) while the convexity of ˝ was proved in Proposition 1.29. To justify assertion (i) of this proposition, observe that the inclusion ri ˝ ˝ is obvious. Fix b ˝, choose a ri ˝, and form the sequence 2 2 1 1 xk a 1 Áb; k N; WD k C k 2
which converges to b as k . Since xk ri ˝ by eorem 1.72(ii), we have b ri ˝. us ! 1 2 2 ˝ ri ˝, which verifies (i). e proof (ii) is similar to that of Proposition 1.31(ii). 1.5 THE DISTANCE FUNCTION e last section of this chapter is devoted to the study of the class of distance functions for con- vex sets, which belongs to the most interesting and important subjects of convex analysis and its extensions. Functions of this type are intrinsically nondifferentiable while they naturally and frequently appear in analysis and applications.
Figure 1.6: Distance function.
Given a set ˝ Rn, the distance function associated with ˝ is defined by d.x ˝/ inf ˚ x ! ˇ ! ˝«: (1.5) I WD k k ˇ 2 Recall further that a mapping f Rn Rp is Lipschitz continuous with constant ` 0 on some W ! set C Rn if we have the estimate f .x/ f .y/ ` x y for all x; y C: (1.6) k k Ä k k 2 Note that the Lipschitz continuity of f in (1.6) specifies its continuity with a linear rate.
Proposition 1.74 Let ˝ be a nonempty subset of Rn. e following hold: 1.5. THE DISTANCE FUNCTION 29 (i) d.x ˝/ 0 if and only if x ˝. I D 2 (ii) e function d.x ˝/ is Lipschitz continuous with constant ` 1 on Rn. I D
Proof. (i) Suppose that d.x ˝/ 0. For each k N, find !k ˝ such that I D 2 2 1 1 0 d.x ˝/ x !k < d.x ˝/ ; D I Ä k k I C k D k which ensures that the sequence !k converges to x, and hence x ˝. f g 2 Conversely, let x ˝ and find a sequence !k ˝ converging to x. en 2 f g
0 d.x ˝/ x !k for all k N; Ä I Ä k k 2 which implies that d.x ˝/ 0 since x !k 0 as k . I D k k ! ! 1 (ii) For any ! ˝, we have the estimate 2 d.x ˝/ x ! x y y ! ; I Ä k k Ä k k C k k which implies in turn that
d.x ˝/ x y inf ˚ y ! ˇ ! ˝« x y d.y ˝/: I Ä k k C k k ˇ 2 D k k C I Similarly, we get d.y ˝/ y x d.x ˝/ and thus d.x ˝/ d.y ˝/ x y , which I Ä k k C I j I I j Ä k k justifies by (1.6) the Lipschitz continuity of d.x ˝/ on Rn with constant ` 1. I D For each x Rn, the Euclidean projection from x to ˝ is defined by 2 ˘.x ˝/ ˚! ˝ ˇ x ! d.x ˝/«: (1.7) I WD 2 ˇ k k D I Proposition 1.75 Let ˝ be a nonempty, closed subset of Rn. en for any x Rn the Euclidean 2 projection ˘.x ˝/ is nonempty. I
Proof. By definition (1.7), for each k N there exists !k ˝ such that 2 2 1 d.x ˝/ x !k < d.x ˝/ : I Ä k k I C k
It is clear that !k is a bounded sequence. us it has a subsequence !k that converges to !. f g f ` g Since ˝ is closed, ! ˝. Letting ` in the inequality 2 ! 1 1 d.x ˝/ x !k` < d.x ˝/ ; I Ä k k I C k` we have d.x ˝/ x ! , which ensures that ! ˘.x ˝/. I D k k 2 I 30 1. CONVEX SETS AND FUNCTIONS An interesting consequence of convexity is the following unique projection property.
Corollary 1.76 If ˝ is a nonempty, closed, convex subset of Rn, then for each x Rn the Euclidean 2 projection ˘.x ˝/ is a singleton. I
Proof. e nonemptiness of the projection ˘.x ˝/ follows from Proposition 1.75. To prove the I uniqueness, suppose that !1;!2 ˘.x ˝/ with !1 !2. en 2 I ¤
x !1 x !2 d.x ˝/: k k D k k D I By the classical parallelogram equality, we have that
2 2 2 2 2 !1 !2 !1 !2 2 x !1 x !1 x !2 2 x C k k : k k D k k C k k D 2 C 2 is directly implies that
2 2 !1 !2 2 !1 !2 2 2 x C x !1 k k < x !1 d.x ˝/ ; 2 D k k 4 k k D I
!1 !2 which is a contradiction due to the inclusion C ˝. 2 2 Now we show that the convexity of a nonempty, closed set and its distance function are equivalent. It is an easy exercise to show that the convexity of an arbitrary set ˝ implies the convexity of its distance function.
Proposition 1.77 Let ˝ be a nonempty, closed subset of Rn. en the function d. ˝/ is convex if I and only if the set ˝ is convex.
n Proof. Suppose that ˝ is convex. Taking x1; x2 R and !i ˘.xi ˝/, we have 2 WD I
xi !i d.xi ˝/ for i 1; 2: k k D I D
e convexity of ˝ ensures that !1 .1 /!2 ˝ for any .0; 1/. It yields C 2 2
d x1 .1 /x2 ˝ x1 .1 /x2 Œ!1 .1 /!2 C I Ä k C C k x1 !1 .1 / x2 !2 Ä k k C k k d.x1 ˝1/ .1 /d.x2 ˝2/; D I C I which implies therefore the convexity of the distance function d. ˝/ by I
d x1 .1 /x2 ˝ d.x1 ˝/ .1 /d.x2 ˝/: C I Ä I C I 1.5. THE DISTANCE FUNCTION 31
To prove the converse implication, suppose that d. ˝/ is convex and fix any !i ˝ for i 1; 2 I 2 D and .0; 1/. en we have 2
d !1 .1 /!2 ˝ d.!1 ˝/ .1 /d.!2 ˝/ 0: C I Ä I C I D
Since ˝ is closed, this yields !1 .1 /!2 ˝ and so justifies the convexity of ˝. C 2 Next we characterize the Euclidean projection to convex sets in Rn. In the proposition below and in what follows we often identify the projection ˘.x ˝/ with its unique element if ˝ I is a nonempty, closed, convex set.
Figure 1.7: Euclidean projection.
Proposition 1.78 Let ˝ be a nonempty, convex subset of Rn and let ! ˝. en we have ! N 2 N 2 ˘.x ˝/ if and only if NI x !;! ! 0 for all ! ˝: (1.8) h N N N i Ä 2
Proof. Take ! ˘.x ˝/ and get for any ! ˝, .0; 1/ that ! .! !/ ˝. us N 2 NI 2 2 N C N 2 2 x ! 2 d.x ˝/ x ! .! !/ 2 k N N k D NI Ä k N N C N k x ! 2 2 x !;! ! 2 ! ! 2: D k N N k h N N N i C k N k is readily implies that 2 x !;! ! ! ! 2: h N N N i Ä k N k Letting 0 , we arrive at (1.8). ! C 32 1. CONVEX SETS AND FUNCTIONS To verify the converse, suppose that (1.8) holds. en for any ! ˝ we get 2 x ! 2 x ! ! ! 2 k N k D k N N CN k x ! 2 ! ! 2 2 x !; ! ! D k N N k C k N k C h N N N i x ! 2 ! ! 2 2 x !;! ! x ! 2: D k N N k C k N k h N N N i k N N k us x ! x ! for all ! ˝, which implies ! ˘.x ˝/ and completes the proof. k N N k Ä k N k 2 N 2 NI
We know from the above that for any nonempty, closed, convex set ˝ in Rn the Euclidean projection ˘.x ˝/ is a singleton. Now we show that the projection mapping is in fact nonexpan- I sive, i.e., it satisfies the following Lipschitz property.
n Proposition 1.79 Let ˝ be a nonempty, closed, convex subset of R . en for any elements x1; x2 2 Rn we have the estimate
2 ˘.x1 ˝/ ˘.x2 ˝/ ˝˘.x1 ˝/ ˘.x2 ˝/; x1 x2˛: I I Ä I I In particular, it implies the Lipschitz continuity of the projection with constant ` 1: D n ˘.x1 ˝/ ˘.x2 ˝/ x1 x2 for all x1; x2 R : I I Ä k k 2
Proof. It follows from the preceding proposition that
n ˝˘.x2 ˝/ ˘.x1 ˝/; x1 ˘.x1 ˝/˛ 0 for all x1; x2 R : I I I Ä 2
Changing the roles of x1; x2 in the inequality above and summing them up give us
˝˘.x1 ˝/ ˘.x2 ˝/; x2 x1 ˘.x1 ˝/ ˘.x2 ˝/˛ 0: I I C I I Ä is implies the first estimate in the proposition. Finally, the nonexpansive property of the Eu- clidean projection follows directly from
2 ˘.x1 ˝/ ˘.x2 ˝/ ˝˘.x1 ˝/ ˘.x2 ˝/; x1 x2˛ I I Ä I I ˘.x1 ˝/ ˘.x2 ˝/ x1 x2 Ä k I I k k k n for all x1; x2 R , which completes the proof of the proposition. 2 1.6 EXERCISES FOR CHAPTER 1
n Exercise 1.1 Let ˝1 and ˝2 be nonempty, closed, convex subsets of R such that ˝1 is bounded or ˝2 is bounded. Show that ˝1 ˝2 is a nonempty, closed, convex set. Give an example of two nonempty, closed, convex sets ˝1; ˝2 for which ˝1 ˝2 is not closed. 1.6. EXERCISES FOR CHAPTER 1 33 Exercise 1.2 Let ˝ be a subset of Rn. We say that ˝ is a cone if x ˝ whenever 0 and 2 x ˝. Show that the following are equivalent: 2 (i) ˝ is a convex cone. (ii) x y ˝ whenever x; y ˝, and x ˝ whenever x ˝ and 0. C 2 2 2 2
Exercise 1.3 (i) Let ˝ be a nonempty, convex set that contains 0 and let 0 1 2. Show Ä Ä that 1˝ 2˝: (ii) Let ˝ be a nonempty, convex set and let ˛; ˇ 0. Show that ˛˝ ˇ˝ .˛ ˇ/˝. C C n Exercise 1.4 (i) Let ˝i for i 1; : : : ; m be nonempty, convex sets in R . Show that m D x co Si 1 ˝i if and only if there exist elements !i ˝i and i 0 for i 1; : : : ; m with 2m D m 2 D Pi 1 i 1 such that x Pi 1 i !i . D D D D n (ii) Let ˝i for i 1; : : : ; m be nonempty, convex cones in R . Show that D m m
X ˝i co [ ˝i : D f g i 1 i 1 D D
Exercise 1.5 Let ˝ be a nonempty, convex cone in Rn. Show that ˝ is a linear subspace of Rn if and only if ˝ ˝. D Exercise 1.6 Show that the following functions are convex on Rn: (i) f .x/ ˛ x , where ˛ 0. D k k (ii) f .x/ x a 2, where a Rn. D k k 2 (iii) f .x/ Ax b , where A is an p n matrix and b Rp. D k k 2 (iv) f .x/ x q, where q 1. D k k Exercise 1.7 Show that the following functions are convex on the given domains: (i) f .x/ eax; x R, where a is a constant. D 2 (ii) f .x/ xq, x Œ0; /, where q 1 is a constant. D 2 1 (iii) f .x/ ln.x/, x .0; /. D 2 1 (iv) f .x/ x ln.x/, x .0; /. D 2 1 Exercise 1.8 (i) Give an example of a function f Rn Rp R, which is convex with respect W ! to each variable but not convex with respect to both. n (ii) Let fi R R for i 1; 2 be convex functions. Can we conclude that the minimum function W ! D min f1; f2 .x/ min f1.x/; f2.x/ is convex? f g WD f g Exercise 1.9 Give an example showing that the product of two real-valued convex functions is not necessarily convex. 34 1. CONVEX SETS AND FUNCTIONS Exercise 1.10 Verify that the set .x; t/ Rn R t x is closed and convex. f 2 j k kg Exercise 1.11 e indicator function associated with a set ˝ is defined by
(0 if x ˝; ı.x ˝/ 2 I WD otherwise: 1 (i) Calculate ı. ˝/ for ˝ , ˝ Rn, and ˝ Œ 1; 1. I D; D D (ii) Show that the set ˝ is convex set if and only if its indicator function ı. ˝/ is convex. I Exercise 1.12 Show that if f R Œ0; / is a convex function, then its q-power f q.x/ W ! 1 WD .f .x//q is also convex for any q 1. n Exercise 1.13 Let fi R R for i 1; 2 be convex functions. Define W ! D n ˝1 .x; 1; 2/ R R R 1 f1.x/ ; WD f 2 n j g ˝2 ˚.x; 1; 2/ R R R 2 f2.x/ : WD 2 j g (i) Show that the sets ˝1; ˝2 are convex. n 2 (ii) Define the set-valued mapping F R R by F.x/ Œf1.x/; / Œf2.x/; / and verify W ! WD 1 1 that the graph of F is ˝1 ˝2. \ Exercise 1.14 We say that f Rn R is positively homogeneous if f .˛x/ ˛f.x/ for all ˛ > W ! D 0, and that f is subadditive if f .x y/ f .x/ f .y/ for all x; y Rn. Show that a positively C Ä C 2 homogeneous function is convex if and only if it is subadditive.
Exercise 1.15 Given a nonempty set ˝ Rn, define K˝ ˚x ˇ 0; x ˝« [ ˝: WD ˇ 2 D 0 (i) Show that K˝ is a cone. (ii) Show that K˝ is the smallest cone containing ˝. (iii) Show that if ˝ is convex, then the cone K˝ is convex as well.
Exercise 1.16 Let ' Rn Rp R be a convex function and let K be a nonempty, convex W ! subset of Rp. Suppose that for each x Rn the function '.x; / is bounded below on K. Verify 2 that the function on Rn defined by f .x/ inf '.x; y/ y K is convex. WD f j 2 g Exercise 1.17 Let f Rn R be a convex function. W ! (i) Show that for every ˛ R the level set x Rn f .x/ ˛ is convex. 2 f 2 j Ä g (ii) Let ˝ R be a convex set. Is it true in general that the inverse image f 1.˝/ is a convex subset of Rn? 1.6. EXERCISES FOR CHAPTER 1 35 n 1 Exercise 1.18 Let C be a convex subset of R C . (i) For x Rn, define F .x/ R .x; / C . Show that F is a set-valued mapping with 2 WD f 2 j 2 g convex graph and give an explicit formula for its graph. (ii) For x Rn, define the function 2
fC .x/ inf ˚ R ˇ .x; / C «: (1.9) WD 2 ˇ 2 2 Find an explicit formula for fC when C is the closed unit ball of R . n (iii) For the function fC defined in (ii), show that if fC .x/ > for all x R , then fC is a 1 2 convex function.
Exercise 1.19 Let f Rn R be bounded from below. Define its convexification W ! m m m
.co f /.x/ inf n X i f .xi / ˇ i 0; X i 1; X i xi x; m No: (1.10) WD ˇ D D 2 i 1 ˇ i 1 i 1 D D D
Verify that (1.10) is convex with co f fC , C co .epi f /, and fC defined in (1.9). D WD Exercise 1.20 We say that f Rn R is quasiconvex if W ! f x .1 /y/ max ˚f .x/; f .y/« for all x; y Rn; .0; 1/: C Ä 2 2 (i) Show that a function f is quasiconvex if and only if for any ˛ R the level set x 2 f 2 Rn f .x/ ˛ is a convex set. j Ä g (ii) Show that any convex function f Rn R is quasiconvex. Give an example demonstrating W ! that the converse is not true.
Exercise 1.21 Let a be a nonzero element in Rn and let b R. Show that 2 ˝ ˚x Rn ˇ a; x b« WD 2 ˇ h i D is an affine set with dim ˝ 1. D
Exercise 1.22 Let the set v1; : : : ; vm consist of affinely independent elements and let v f g … aff v1; : : : ; vm . Show that v1; : : : ; vm; v are affinely independent. f g Exercise 1.23 Suppose that ˝ is a convex subset of Rn with dim ˝ m, m 1. Let the set D v1; : : : ; vm ˝ consist of affinely independent elements and let f g
m co ˚v1; : : : ; vm«: WD
Show that aff ˝ aff m aff ˚v1; : : : ; vm«. D D 36 1. CONVEX SETS AND FUNCTIONS Exercise 1.24 Let ˝ be a nonempty, convex subset of Rn. (i) Show that aff ˝ aff ˝. D (ii) Show that for any x ri ˝ and x ˝ there exists t > 0 such that x t.x x/ ˝. N 2 2 N C N 2 (iii) Prove Proposition 1.73(ii).
n Exercise 1.25 Let ˝1; ˝2 R be convex sets with ri ˝1 ri ˝2 . Show that: \ ¤ ; (i) ˝1 ˝2 ˝1 ˝2. \ D \ (ii) ri .˝1 ˝2/ ri ˝1 ri˝2. \ D \ n Exercise 1.26 Let ˝1; ˝2 R be convex sets with ˝1 ˝2. Show that ri ˝1 ri ˝2. D D Exercise 1.27 (i) Let B Rn Rp be an affine mapping and let ˝ be a convex subset of Rn. W ! Prove the equality B.ri ˝/ ri B.˝/: D n (ii) Let ˝1 and ˝2 be convex subsets of R . Show that ri .˝1 ˝2/ ri ˝1 ri ˝2. D Exercise 1.28 Let f Rn R be a convex function. Show that: W ! (i) aff .epi f / aff .dom f / R : D (ii) ri .epi f / ˚.x; / ˇ x ri .dom f /; > f .x/«: D ˇ 2 Exercise 1.29 Find the explicit formulas for the distance function d.x ˝/ and the Euclidean I projection ˘.x ˝/ in the following cases: I (i) ˝ is the closed unit ball of Rn. 2 (ii) ˝ ˚.x1; x2/ R ˇ x1 x2 1«. WD 2 ˇ j j C j j Ä (iii) ˝ Œ 1; 1 Œ 1; 1. WD Exercise 1.30 Find the formulas for the projection ˘.x ˝/ in the following cases: I (i) ˝ ˚x Rn ˇ a; x b«, where 0 a Rn and b R : WD 2 ˇ h i D ¤ 2 2 (ii) ˝ ˚x Rn ˇ Ax b«, where A is an m n matrix with rank A m and b Rm. WD 2 ˇ D D 2 (iii) ˝ is the nonnegative orthant ˝ Rn . WD C
Exercise 1.31 For ai bi with i 1; : : : ; n, define Ä D n ˝ ˚x .x1; : : : ; xn/ R ˇ ai x bi for all i 1; : : : ; n«: WD D 2 ˇ Ä Ä D Show that the projection ˘.x ˝/ has the following representation: I n ˘.x ˝/ ˚u .u1; : : : ; un/ R ˇ ui max ai ; min b; xi for i 1; : : : ; n «: I D D 2 ˇ D f f gg 2 f g 37 C H A P T E R 2 Subdifferential Calculus
In this chapter we present basic results of generalized differentiation theory for convex functions and sets. A geometric approach of variational analysis based on convex separation for extremal systems of sets allows us to derive general calculus rules for normals to sets and subgradients of functions. We also include into this chapter some related results on Lipschitz continuity of convex functions and convex duality. Note that some extended versions of the presented calculus results can be found in exercises for this chapter, with hints to their proofs given at the end of the book.
2.1 CONVEX SEPARATION We start this section with convex separation theorems, which play a fundamental role in convex analysis and particularly in deriving calculus rules by the geometric approach. e first result concerns the strict separation of a nonempty, closed, convex set and an out-of-set point.
Proposition 2.1 Let ˝ be a nonempty, closed, convex subset of Rn and let x ˝. en there is a N … nonzero element v Rn such that 2 sup ˚ v; x ˇ x ˝« < v; x : (2.1) h i ˇ 2 h Ni
Proof. Denote w ˘.x ˝/ and let v x w 0. Applying Proposition 1.78 gives us N WD NI WD N N ¤ v; x w x w; x w 0 for any x ˝: h N i D h N N N i Ä 2 It follows furthermore that
v; x .x v/ v; x w 0; h N i D h N i Ä which implies in turn that v; x v; x v 2 and thus verifies (2.1). h i Ä h Ni k k In the setting of Proposition 2.1 we choose a number b R such that 2 sup ˚ v; x ˇ x ˝« < b < v; x h i ˇ 2 h Ni and define the function A.x/ v; x for which A.x/ > b and A.x/ < b for all x ˝. en we WD h i N 2 say that x and ˝ are strictly separated by the hyperplane L x Rn A.x/ b . N WD f 2 j D g e next theorem extends the result of Proposition 2.1 to the case of two nonempty, closed, convex sets at least one of which is bounded. 38 2. SUBDIFFERENTIAL CALCULUS n eorem 2.2 Let ˝1 and ˝2 be nonempty, closed, convex subsets of R with ˝1 ˝2 . If n \ D; ˝1 is bounded or ˝2 is bounded, then there is a nonzero element v R such that 2
sup ˚ v; x ˇ x ˝1« < inf ˚ v; y ˇ y ˝2«: h i ˇ 2 h i ˇ 2
Proof. Denote ˝ ˝1 ˝2. en ˝ is a nonempty, closed, convex set and 0 ˝. Applying WD … Proposition 2.1 to ˝ and x 0, we have N D sup ˚ v; x ˇ x ˝« < 0 v; x with some 0 v Rn : WD h i ˇ 2 D h Ni ¤ 2
For any x ˝1 and y ˝2, we have v; x y , and so v; x v; y . erefore, 2 2 h i Ä h i Ä C h i
sup ˚ v; x ˇ x ˝1« inf ˚ v; y ˇ y ˝2« < inf ˚ v; y ˇ y ˝2«; h i ˇ 2 Ä C h i ˇ 2 h i ˇ 2 which completes the proof of the theorem.
Figure 2.1: Illustration of convex separation.
Next we show that two nonempty, closed, convex sets with empty intersection in Rn can be separated, while not strictly in general, if both sets are unbounded.
n Corollary 2.3 Let ˝1 and ˝2 be nonempty, closed, convex subsets of R such that ˝1 ˝2 . \ D; en they are in the sense that there is a nonzero element v Rn with 2
sup ˚ v; x ˇ x ˝1« inf ˚ v; y ˇ y ˝2«: (2.2) h i ˇ 2 Ä h i ˇ 2 2.1. CONVEX SEPARATION 39
Proof. For every fixed k N, we define the set k ˝2 IB.0 k/ and observe that it is a 2 WD \ I closed, bounded, convex set, which is also nonempty if k is sufficiently large. Employing eo- n rem 2.2 allows us to find 0 uk R such that ¤ 2
uk; x < uk; y for all x ˝1; y k: h i h i 2 2 uk Denote vk and assume without loss of generality that vk v 0 as k . Fix fur- WD uk ! ¤ ! 1 k k ther x ˝1 and y ˝2 and take k0 N such that y k for all k k0. en 2 2 2 2
vk; x < vk; y whenever k k0: h i h i Letting k , we have v; x v; y , which justifies the claim of the corollary. ! 1 h i Ä h i Now our intention is to establish a general separation theorem for convex sets that may not be closed. To proceed, we verify first the following useful result.
Figure 2.2: Geometric illustration of Lemma 2.4.
Lemma 2.4 Let ˝ be a nonempty, convex subset of Rn such that 0 ˝ ˝. en there is a sequence 2 n xk 0 as k with xk ˝ for all k N. ! ! 1 … 2
Proof. Recall that ri ˝ by eorem 1.72(i). Fix x0 ri ˝ and show that tx0 ˝ for all ¤ ; 2 … t > 0; see Figure 2.2. Indeed, suppose the contrary that tx0 ˝ and get by eorem 1.72(ii) 2 that t 1 0 x0 . tx0/ ri ˝ ˝; D 1 t C 1 t 2 C C x0 a contradiction. Letting then xk gives us xk ˝ for every k and xk 0. WD k … ! 40 2. SUBDIFFERENTIAL CALCULUS Note that Lemma 2.4 may not hold for nonconvex sets. As an example, consider the set ˝ Rn 0 for which 0 ˝ ˝ while the conclusion of the Lemma 2.4 fails. WD nf g 2 n
n eorem 2.5 Let ˝1 and ˝2 be nonempty, convex subsets of R with ˝1 ˝2 . en they \ D; are separated in the sense of (2.2) with some v 0. ¤
Proof. Consider the set ˝ ˝1 ˝2 for which 0 ˝. We split the proof of the theorem into WD … the following two cases: Case 1: 0 ˝. In this case we apply Proposition 2.1 and find v 0 such that … ¤ v; z < 0 for all z ˝: h i 2
is gives us v; x < v; y whenever x ˝1 and y ˝2, which justifies the statement. h i h i 2 2 Case 2: 0 ˝. In this case we employ Lemma 2.4 and find a sequence xk 0 with xk ˝. 2 ! … en Proposition 2.1 produces a sequence vk of nonzero elements such that f g
vk; z < vk; xk for all z ˝: h i h i 2
Without loss of generality, we get (dividing by vk and extracting a convergent subsequence) k k that vk v and vk v 1. is gives us by letting k that ! k k D k k D ! 1 v; z 0 for all z ˝: h i Ä 2 To complete the proof, we just repeat the arguments of Case 1.
Figure 2.3: Extremal system of sets.
eorem 2.5 ensures the separation of convex sets, which do not intersect. To proceed now with separation of convex sets which may have common points as in Figure 2.3, we introduce the notion of set extremality borrowed from variational analysis that deals with both convex and nonconvex sets while emphasizing intrinsic optimality/extremality structures of problems under consideration; see, e.g., [14] for more details. 2.1. CONVEX SEPARATION 41 n Definition 2.6 Given two nonempty, convex sets ˝1; ˝2 R , we say that the set system ˝1; ˝2 f g is if for any > 0 there is a Rn such that a < and 2 k k
.˝1 a/ ˝2 : \ D; e next proposition gives us an important example of extremal systems of sets.
Proposition 2.7 Let ˝ be a nonempty, convex subset of Rn and let x ˝ be a boundary point of N 2 ˝. en the sets ˝ and x form an extremal system. f Ng
Proof. Since x bd ˝, for any number > 0 we get IB.x =2/ ˝c . us choosing b N 2 NI \ ¤ ; 2 IB.x =2/ ˝c and denoting a b x yield a < and .˝ a/ x . NI \ WD N k k \ f Ng D ; e following result is a separation theorem for extremal systems of convex sets. It admits a far-going extension to nonconvex settings, which is known as the extremal principle and is at the heart of variational analysis and its applications [14]. It is worth mentioning that, in con- trast to general settings of variational analysis, we do not impose here any closedness assumptions on the sets in question. is will allow us to derive in the subsequent sections of this chapter various calculus rules for normals to convex sets and subgradients of convex functions by using a simple variational device without imposing closedness and lower semicontinuity assumptions conventional in variational analysis.
n eorem 2.8 Let ˝1 and ˝2 be two nonempty, convex subsets of R . en ˝1 and ˝2 form n an extremal system if and only if there is a nonzero element v R which separates the sets ˝1 2 and ˝2 in the sense of (2.2).
n Proof. Suppose that ˝1 and ˝2 form an extremal system of convex sets in R . en there is a n sequence ak in R with ak 0 and f g !
.˝1 ak/ ˝2 ; k N : \ D; 2
For every k N, apply eorem 2.5 to the convex sets ˝1 ak and ˝2 and find a sequence of 2 n nonzero elements uk R satisfying 2
uk; x ak uk; y for all x ˝1 and y ˝2; k N : h i Ä h i 2 2 2 uk e normalization vk with vk 1 gives us WD uk k k D k k vk; x ak vk; y for all x ˝1 and y ˝2: (2.3) h i Ä h i 2 2 n us we find v R with v 1 such that vk v as k along a subsequence. Passing to 2 k k D ! ! 1 the limit in (2.3) verifies the validity of (2.2). 42 2. SUBDIFFERENTIAL CALCULUS For the converse implication, suppose that there is a nonzero element v Rn which sepa- 2 rates the sets ˝1 and ˝2 in the sense of (2.2). en it is not hard to see that 1 ˝1 vÁ ˝2 for every k N : k \ D; 2
us ˝1 and ˝2 form an extremal system.
2.2 NORMALS TO CONVEX SETS In this section we apply the separation results obtained above (particularly, the extremal version in eorem 2.8) to the study of generalized normals to convex sets and derive basic rules of the normal cone calculus.
Definition 2.9 Let ˝ Rn be a convex set with x ˝. e to ˝ at x is N 2 N N.x ˝/ ˚v Rnˇ v; x x 0 for all x ˝«: (2.4) NI WD 2 ˇ h Ni Ä 2 By convention, we put N.x ˝/ for x ˝. NI WD ; N …
Figure 2.4: Epigraph of the absolute value function and the normal cone to it at the origin.
We first list some elementary properties of the normal cone defined above.
Proposition 2.10 Let x ˝ for a convex subset ˝ of Rn. en we have: N 2 (i) N.x ˝/ is a closed, convex cone containing the origin. NI (ii) If x int ˝, then N.x ˝/ 0 . N 2 NI D f g
Proof. To verify (i), fix vi N.x ˝/ and i 0 for i 1; 2. We get by the definition that 2 NI D 0 N.x ˝/ and vi ; x x 0 for all x ˝ implying 2 NI h Ni Ä 2
1v1 2v2; x x 0 whenever x ˝: h C Ni Ä 2 2.2. NORMALS TO CONVEX SETS 43
us 1v1 2v2 N.x ˝/, and hence N.x ˝/ is a convex cone. To check its closedness, fix C 2 NI NI a sequence vk N.x ˝/ that converges to v. en passing to the limit in f g NI
vk; x x 0 for all x ˝ as k h Ni Ä 2 ! 1 yields v; x x 0 whenever x ˝, and so v N.x ˝/. is completes the proof of (i). h Ni Ä 2 2 NI To check (ii), pick v N.x ˝/ with x int ˝ and find ı > 0 such that x ıIB ˝. 2 NI N 2 N C us for every e IB we have the inequality 2 v; x ıe x 0; h N C Ni Ä which shows that ı v; e 0, and hence v; e 0. Choose t > 0 so small that tv IB. en h i Ä h i Ä 2 v; tv 0 and therefore t v 2 0, i.e., v 0. h i Ä k k D D We start calculus rules with deriving rather simple while very useful results.
n p Proposition 2.11 (i) Let ˝1 and ˝2 be nonempty, convex subsets of R and R , respectively. For .x1; x2/ ˝1 ˝2, we have N N 2
N .x1; x2/ ˝1 ˝2 N.x1 ˝1/ N.x2 ˝2/: N N I D N I N I n (ii) Let ˝1 and ˝2 be convex subsets of R with xi ˝i for i 1; 2. en N 2 D
N.x1 x2 ˝1 ˝2/ N.x1 ˝1/ N.x2 ˝2/: N CN I C D N I \ N I
Proof. To verify (i), fix .v1; v2/ N..x1; x2/ ˝1 ˝2/ and get by the definition that 2 N N I
.v1; v2/; .x1; x2/ .x1; x2/ v1; x1 x1 v2; x2 x2 0 (2.5) h N N i D h N i C h N i Ä whenever .x1; x2/ ˝1 ˝2. Putting x2 x2 in (2.5) gives us 2 WD N
v1; x1 x1 0 for all x1 ˝1; h N i Ä 2 which means that v1 N.x1 ˝1/. Similarly, we obtain v2 N.x2 ˝2/ and thus justify the in- 2 N I 2 N I clusion “ ” in (2.5). e opposite inclusion is obvious. Let us now verify (ii). Fix v N.x1 x2 ˝1 ˝2/ and get by the definition that 2 N CN I C
v; x1 x2 .x1 x2/ 0 whenever x1 ˝1; x2 ˝2: h C N CN i Ä 2 2
Putting there x1 x1 and x2 x2 gives us v N.x1 ˝1/ N.x2 ˝2/. e opposite inclusion WD N WD N 2 N I \ N I in (ii) is also straightforward. 44 2. SUBDIFFERENTIAL CALCULUS Consider further a linear mapping A Rn Rp and associate it with a p n matrix as W ! usual. e adjoint mapping A Rp Rn is defined by W ! n p Ax; y x; Ay for all x R ; y R ; h i D h i 2 2 which corresponds to the matrix transposition. e following calculus result describes normals to solution sets for systems of linear equations.
Proposition 2.12 Let B Rn Rp be defined by B.x/ Ax b, where A Rn Rp is a linear W ! WD C W ! mapping and b Rp. Given c R, consider the solution set 2 2 ˝ ˚x Rn ˇ Bx c«: WD 2 ˇ D For any x ˝ with B.x/ c, we have N 2 N D n p N.x ˝/ im A ˚v R ˇ v Ay; y R «: (2.6) NI D D 2 ˇ D 2
Proof. Take v N.x ˝/ and get v; x x 0 for all x such that Ax b c. Fixing any 2 NI h Ni Ä C D u ker A, we see that A.x u/ Ax c b, i.e., A.x u/ b c, which yields v; u 0 2 N D N D N C D h i whenever u ker A. Since u ker A, it follows that v; u 0 for all u ker A. Arguing by 2 2 h i D 2 contradiction, suppose that there is no y Rp with v A y. us v W A Rp, where the 2 D … WD set W Rn is obviously nonempty, closed, and convex. Employing Proposition 2.1 gives us a nonzero element u Rn satisfying N 2 sup ˚ u; w ˇ w W « < u; v ; h N i ˇ 2 h N i and implying that 0 < u; v since 0 W . Furthermore h N i 2 p t u; Ay u; A.ty/ < u; v for all t R; y R : h N i D h N i h N i 2 2 Hence u; A y 0, which shows that Au; y 0 whenever y Rp, i.e., Au 0. us u h N i D h N i D 2 N D N 2 ker A while v; u > 0. is contradiction justifies the inclusion “ ” in (2.6). h Ni To verify the opposite inclusion in (2.6), fix any v Rn with v A y for some y Rp. 2 D 2 For any x ˝, we have 2
v; x x Ay; x x y; Ax Ax 0; h Ni D h Ni D h Ni D which means that v N.x ˝/ and thus completes the proof of the proposition. 2 NI e following proposition allows us to characterize the separation of convex (generally non- closed) sets in terms of their normal cones.
n Proposition 2.13 Let ˝1 and ˝2 be convex subsets of R with x ˝1 ˝2. en the following N 2 \ assertions are equivalent: 2.2. NORMALS TO CONVEX SETS 45 n (i) ˝1; ˝2 forms an extremal system of two convex sets in R . f g (ii) e sets ˝1 and ˝2 are separated in the sense of (2.2) with some v 0. ¤ (iii) N.x ˝1/ N.x ˝2/ 0 . NI \ NI ¤ f g
Proof. e equivalence of (i) and (ii) follows from eorem 2.8. Suppose that (ii) is satisfied. Since x ˝2, we have N 2 sup ˚ v; x ˇ x ˝1« v; x ; h i ˇ 2 Ä h Ni and thus v; x x 0 for all x ˝1, i.e., v N.x ˝1/. Similarly, we can verify the inclusion h Ni Ä 2 2 NI v N.x ˝2/, and hence arrive at (iii). 2 NI Now suppose that (iii) is satisfied and let 0 v N.x ˝1/ N.x ˝2/. en (2.2) ¤ 2 NI \ NI follows directly from the definition of the normal cone. Indeed, for any x ˝1 and y ˝2 one 2 2 has v; x x 0 and v; y x 0: h Ni Ä h Ni Ä is implies v; x x 0 v; y x , and hence v; x v; y . h Ni Ä Ä h Ni h i Ä h i e following corollary provides a normal cone characterization of boundary points.
Corollary 2.14 Let ˝ be a nonempty, convex subset of Rn and let x ˝. en x bd ˝ if and N 2 N 2 only if the normal cone to ˝ is nontrivial, i.e., N.x ˝/ 0 . NI ¤ f g
Proof. e “only if ” part follows from Proposition 2.10(ii). To justify the “if ” part, take x bd ˝ N 2 and consider the sets ˝1 ˝ and ˝2 x , which form an extremal system by Proposition 2.7. WDn WD f Ng Note that N.x ˝2/ R . us the claimed result is a direct consequence of Proposition 2.13 NI D since N.x ˝1/ Œ N.x ˝2/ N.x ˝1/ 0 . NI \ NI D NI ¤ f g Now we are ready to establish a major result of the normal cone calculus for convex sets providing a relationship between normals to set intersections and normals to each set in the in- tersection. Its proof is also based on convex separation in extremal systems of sets. Note that this approach allows us to capture general convex sets, which may not be closed.
n eorem 2.15 Let ˝1 and ˝2 be convex subsets of R with x ˝1 ˝2. en for any v N 2 \ 2 N.x ˝1 ˝2/ there exist 0 and vi N.x ˝i /, i 1; 2, such that NI \ 2 NI D
v v1 v2; .; v1/ .0; 0/: (2.7) D C ¤
Proof. Fix v N.x ˝1 ˝2/ and get by the definition that 2 NI \
v; x x 0 for all x ˝1 ˝2: h Ni Ä 2 \ 46 2. SUBDIFFERENTIAL CALCULUS
Denote 1 ˝1 Œ0; / and 2 .x; / x ˝2; v; x x . ese convex sets WD 1 WD f j 2 Ä h Nig form an extremal system since we obviously have .x; 0/ 1 2 and N 2 \
1 .2 .0; a// whenever a > 0: \ D; By eorem 2.8, find 0 .w; / Rn R such that ¤ 2
w; x 1 w; y 2 for all .x; 1/ 1; .y; 2/ 2: (2.8) h i C Ä h i C 2 2
It is easy to see that 0. Indeed, assuming > 0 and taking into account that .x; k/ 1 Ä N 2 when k > 0 and .x; 0/ 2, we get N 2 w; x k w; x ; h Ni C Ä h Ni which leads to a contradiction. ere are two possible cases to consider. Case 1: 0. In this case we have w 0 and D ¤
w; x w; y for all x ˝1; y ˝2: h i Ä h i 2 2
en following the proof of Proposition 2.13 gives us w N.x ˝1/ Œ N.x ˝2/, and thus 2 NI \ NI (2.7) holds for 0, v1 w, and v2 w. D D D Case 2: < 0. In this case we let > 0 and deduce from (2.8), by using .x; 0/ 1 when WD 2 x ˝1, and .x; 0/ 2, that 2 N 2
w; x w; x for all x ˝1: h i Ä h Ni 2 w is implies that w N.x ˝1/, and hence N.x ˝1/. To proceed further, we get from (2.8), 2 NI 2 NI due to .x; 0/ 1 and .y; ˛/ 2 for all y ˝2 with ˛ v; y x , that N 2 2 2 WD h Ni
Dw; xE Dw; yE v; y x whenever y ˝2: N Ä C h Ni 2 Dividing both sides by gives us w w D ; xE D ; yE v; y x whenever y ˝2: N C h Ni 2 is clearly implies the inequality w D v; y xE 0 for all y ˝2; C N Ä 2 w w w w and thus v v N.x ˝2/. Letting finally v1 N.x ˝1/ and v2 C D C 2 NI WD 2 NI WD C v N.x ˝2/ gives us v v1 v2, which ensures the validity of (2.7) with 1. 2 NI D C D 2.2. NORMALS TO CONVEX SETS 47
Figure 2.5: Normal cone to set intersection.
Observe that representation (2.7) does not provide a calculus rule for representing normals to set intersections unless 0. However, it is easy to derive such an intersection rule directly from ¤ eorem 2.15 while imposing the following basic qualification condition, which plays a crucial role in the subsequent calculus results; see Figure 2.5.
n Corollary 2.16 Let ˝1 and ˝2 be convex subsets of R with x ˝1 ˝2. Assume that the basic N 2 \ qualification condition (BQC) is satisfied:
N.x ˝1/ N.x ˝2/ 0 : (2.9) NI \ NI D f g en we have the normal cone intersection rule
N.x ˝1 ˝2/ N.x ˝1/ N.x ˝2/: (2.10) NI \ D NI C NI
Proof. For any v N.x ˝1 ˝2/, we find by eorem 2.15 a number 0 and elements vi 2 NI \ 2 N.x ˝i / as i 1; 2 such that the conditions in (2.7) hold. Assuming that 0 in (2.7) gives us NI D D that 0 v1 v2 N.x ˝1/ Œ N.x ˝2/, which contradicts BQC (2.9). us > 0 and ¤ D 2 NI \ NI v v1= v2= N.x ˝1/ N.x ˝2/. Hence the inclusion “ ” in (2.10) is satisfied. e D C 2 NI C NI opposite inclusion in (2.10) can be proved easily using the definition of the normal cone. Indeed, fix any v N.x ˝1/ N.x ˝2/ with v v1 v2, where vi N.x ˝i / for i 1; 2. For any 2 NI C NI D C 2 NI D x ˝1 ˝2, one has 2 \
v; x x v1 v2; x x v1; x x v2; x x 0; h Ni D h C Ni D h Ni C h Ni Ä which implies v N.x ˝1 ˝2/ and completes the proof of the corollary. 2 NI \ 48 2. SUBDIFFERENTIAL CALCULUS e following example shows that the intersection rule (2.10) does not hold generally with- out the validity of the basic qualification condition (2.9).
2 2 2 2 Example 2.17 Let ˝1 .x; / R x and let ˝2 .x; / R x . en WD f 2 j 2 g WD f 2 j Ä g for x .0; 0/ we have N.x ˝1 ˝2/ R while N.x ˝1/ N.x ˝2/ 0 R. N D NI \ D NI C NI D f g Next we present an easily verifiable condition ensuring the validity of BQC (2.9). More general results related to relative interior are presented in Exercise 2.13 and its solution.
n Corollary 2.18 Let ˝1; ˝2 R be such that int ˝1 ˝2 or int ˝2 ˝1 . en for any \ ¤ ; \ ¤ ; x ˝1 ˝2 both BQC (2.9) and intersection rule (2.10) are satisfied. N 2 \
Proof. Consider for definiteness the case where int ˝1 ˝2 . Let us fix x ˝1 ˝2 and \ ¤ ; N 2 \ show that BQC (2.9) holds, which ensures the intersection rule (2.10) by Corollary 2.16. Arguing by contradiction, suppose the opposite
N.x ˝1/ N.x ˝2/ 0 NI \ NI ¤ f g n and then find by Proposition 2.13 a nonzero element v R that separates the sets ˝1 and ˝2 as 2 in (2.2). Fixing now u int ˝1 ˝2 gives us a number ı > 0 such that u ıIB ˝1. It ensures N 2 \ N C by (2.2) that v; x v; u for all x u ıIB: h i Ä h Ni 2 N C is implies in turn the inequality v; u ı v; e v; u whenever e IB h Ni C h i Ä h Ni 2 v and tells us that v; e 0 for all e IB. Choosing finally e , we arrive at a contradiction h i Ä 2 WD v and thus complete the proof of this corollary. k k Employing induction arguments allows us to derive a counterpart of Corollary 2.16 (and, similarly, of Corollary 2.18) for finitely many sets.
n m Corollary 2.19 Let ˝i R for i 1; : : : ; m be convex sets and let x T ˝i . Assume the D N 2 i 1 validity of the following qualification condition: D m h X vi 0; vi N.x ˝i /i vi 0 for all i 1; : : : ; m: D 2 NI H) D D i 1 D en we have the intersection formula m m
N x \ ˝i Á X N.x ˝i /: NI D NI i 1 i 1 D D 2.3. LIPSCHITZ CONTINUITY OF CONVEX FUNCTIONS 49 2.3 LIPSCHITZ CONTINUITY OF CONVEX FUNCTIONS In this section we study the fundamental notion of Lipschitz continuity for convex functions. is notion can be treated as “continuity at a linear rate” being highly important for many aspects of convex and variational analysis and their applications.
Definition 2.20 A function f Rn R is L on a set ˝ dom f if there W ! is a constant ` 0 such that f .x/ f .y/ ` x y for all x; y ˝: (2.11) j j Ä k k 2 We say that f is L around x dom f if there are constants ` 0 N 2 and ı > 0 such that (2.11) holds with ˝ IB.x ı/. D NI Here we give several verifiable conditions that ensure Lipschitz continuity and characterize its localized version for general convex functions. One of the most effective characterizations of local Lipschitz continuity, holding for nonconvex functions as well, is given via the following notion of the singular (or horizon) subdifferential, which was largely underinvestigated in convex analysis and was properly recognized only in the framework of variational analysis; see [14, 27]. Note that, in contrast to the (usual) subdifferential of convex analysis defined in the next section, the singular subdifferential does not reduce to the classical derivative for differentiable functions.
Definition 2.21 Let f Rn R be a convex function and let x dom f . e - W ! N 2 of the function f at x is defined by N n @1f .x/ ˚v R ˇ .v; 0/ N .x; f .x/ epi f «: (2.12) N WD 2 ˇ 2 N N I e following example illustrates the calculation of @ f .x/ based on definition (2.12). 1 N Example 2.22 Consider the function
(0 if x 1; f .x/ j j Ä WD otherwise: 1 en epi f Œ 1; 1 Œ0; / and for x 1 we have D 1 N D N .x; f .x// epi f Œ0; / . ; 0; N N I D 1 1 which shows that @ f .x/ Œ0; /; see Figure 2.6. 1 N D 1 e next proposition significantly simplifies the calculation of @ f .x/. 1 N Proposition 2.23 For a convex function f Rn R with x dom f , we have W ! N 2 @1f .x/ N.x dom f /: N D NI 50 2. SUBDIFFERENTIAL CALCULUS
Figure 2.6: Singular subgradients.
Proof. Fix any v @ f .x/ with x dom f and observe that .x; f .x// epi f . en using 2 1 N 2 2 (2.12) and the normal cone construction (2.4) give us
v; x x v; x x 0 f .x/ f .x/ 0; h Ni D h Ni C N Ä which shows that v N.x dom f /. Conversely, suppose that v N.x dom f / and fix any 2 NI 2 NI .x; / epi f . en we have f .x/ , and so x dom f . us 2 Ä 2 v; x x 0. f .x// v; x x 0; h Ni C N D h Ni Ä which implies that .v; 0/ N..x; f .x// epi f /, i.e., v @ f .x/. 2 N N I 2 1 N To proceed with the study of Lipschitz continuity of convex functions, we need the follow- ing two lemmas, which are of their own interest.
n Lemma 2.24 Let ei i 1; : : : ; n be the standard orthonormal basis of R . Denote f j D g
A ˚x ei ˇ i 1; : : : ; n«; > 0: WD N ˙ ˇ D en the following properties hold:
(i) x ei co A for and i 1; : : : ; n. N C 2 j j Ä D (ii) IB.x =n/ co A. NI
Proof. (i) For , find t Œ0; 1 with t. / .1 t/. en x ei A gives us j j Ä 2 D C N ˙ 2
x ei t.x ei / .1 t/.x ei / co A: N C D N C N C 2 2.3. LIPSCHITZ CONTINUITY OF CONVEX FUNCTIONS 51
(ii) For x IB.x =n/, we have x x .=n/u with u 1. Represent u via ei by 2 NI DN C k k Ä f g n
u X i ei ; D i 1 D q n 2 where i Pi 1 i u 1 for every i. is gives us j j Ä D D k k Ä n n i 1 x x u x X ei X x i ei Á: DN C n DN C n D n N C i 1 i 1 D D
Denoting i i yields i . It follows from (i) that x i ei x i ei co A, and WD j j Ä N C DN C 2 thus x co A since it is a convex combination of elements in co A. 2
Lemma 2.25 If a convex function f Rn R is bounded above on IB.x ı/ for some element x W ! NI N 2 dom f and number ı > 0, then f is bounded on IB.x ı/. NI
Proof. Denote m f .x/ and take M > 0 with f .x/ M for all x IB.x ı/. Picking any u WD N Ä 2 NI 2 IB.x ı/, consider the element x 2x u. en x IB.x ı/ and NI WD N 2 NI x u 1 1 m f .x/ f C f .x/ f .u/; D N D 2 Ä 2 C 2 which shows that f .u/ 2m f .x/ 2m M and thus f is bounded on IB.x ı/. NI e next major result shows that the local boundedness of a convex function implies its local Lipschitz continuity.
eorem 2.26 Let f Rn R be convex with x dom f . If f is bounded above on IB.x ı/ W ! N 2 NI for some ı > 0, then f is Lipschitz continuous on IB.x ı=2/. NI
Proof. Fix x; y IB.x ı=2/ with x y and consider the element 2 NI ¤ ı u x x yÁ: WD C 2 x y k k x y ı ı ı Since e IB, we have u x e x IB IB x ıIB. Denoting further WD x y 2 D C 2 2 N C 2 C 2 N C ı k k ˛ gives us u x ˛.x y/, and thus WD 2 x y D C k k 1 ˛ x u y: D ˛ 1 C ˛ 1 C C 52 2. SUBDIFFERENTIAL CALCULUS It follows from the convexity of f that 1 ˛ f .x/ f .u/ f .y/; Ä ˛ 1 C ˛ 1 C C which implies in turn the inequalities
1 1 2 x y 4M f .x/ f .y/ .f .u/ f .y// 2M 2M k k x y Ä ˛ 1 Ä ˛ 1 D ı 2 x y Ä ı k k C C C k k with M sup f .x/ x IB.x ı/ < by Lemma 2.25. Interchanging the role of x and y WD fj j j 2 NI g 1 above, we arrive at the estimate 4M f .x/ f .y/ x y j j Ä ı k k and thus verify the Lipschitz continuity of f on IB.x ı=2/. NI e following two consequences of eorem 2.26 ensure the automatic local Lipschitz con- tinuity of a convex function on appropriate sets.
Corollary 2.27 Any extended-real-valued convex function f Rn R is locally Lipschitz contin- W ! uous on in the interior of its domain int.dom f /.
Proof. Pick x int.dom f / and choose > 0 such that x ei dom f for every i. Consider- N 2 N ˙ 2 ing the set A from Lemma 2.24, we get IB.x =n/ co A by assertion (ii) of this lemma. Denote NI M max f .a/ a A < over the finite set A. Using the representation WD f j 2 g 1 m m
x X i ai with i 0 X i 1; ai A D D 2 i 1 i 1 D D for any x IB.x =n/ shows that 2 NI m m
f .x/ X i f .ai / X i M M; Ä Ä D i 1 i 1 D D and so f is bounded above on IB.x =n/. en eorem 2.26 tells us that f is Lipschitz con- NI tinuous on IB.x =2n/ and thus it is locally Lipschitz continuous on int .dom f /. NI It immediately follows from Corollary 2.27 that any finite convex function on Rn is always locally Lipschitz continuous on the whole space.
Corollary 2.28 If f Rn R is convex, then it is locally Lipschitz continuous on Rn. W ! 2.4. SUBGRADIENTS OF CONVEX FUNCTIONS 53 e final result of this section provides several characterizations of the local Lipschitz con- tinuity of extended-real-valued convex functions.
eorem 2.29 Let f Rn R be convex with x dom f . e following are equivalent: W ! N 2 (i) f is continuous at x. N (ii) x int.dom f /. N 2 (iii) f is locally Lipschitz continuous around x. N (iv) @ f .x/ 0 . 1 N D f g
Proof. To verify (i) (ii), by the continuity of f at x, for any > 0 find ı > 0 such that H) N f .x/ f .x/ < whenever x IB.x ı/: j N j 2 NI Since f .x/ < , this implies that IB.x ı/ dom f , and so x int.dom f /. N 1 NI N 2 Implication (ii) (iii) follows directly from Corollary 2.27 while (iii) (i) is obvious. us H) H) conditions (i), (ii), and (iii) are equivalent. To show next that (ii) (iv), take x int.dom f / H) N 2 for which N.x dom f / 0 . en Proposition 2.23 yields @ f .x/ 0 . NI D f g 1 N D f g To verify finally (iv) (ii), we argue by contradiction and suppose that x int.dom f /. H) N … en x bd.dom f / and it follows from Corollary 2.14 that N.x dom f / 0 , which contra- N 2 NI ¤ f g dicts the condition @ f .x/ 0 by Proposition 2.23. 1 N D f g 2.4 SUBGRADIENTS OF CONVEX FUNCTIONS In this section we define and start studying the concept of subdifferential (or the collection of sub- gradients) for convex extended-real-valued functions. is notion of generalized derivative is one of the most fundamental in analysis, with a variety of important applications. e revolutionary idea behind the subdifferential concept, which distinguishes it from other notions of generalized derivatives in mathematics, is its set-valuedness as the indication of nonsmoothness of a function around the reference individual point. It not only provides various advantages and flexibility but also creates certain difficulties in developing its calculus and applications. e major techniques of subdifferential analysis revolve around convex separation of sets.
Definition 2.30 Let f Rn R be a convex function and let x dom f . An element v Rn is W ! N 2 2 called a of f at x if N v; x x f .x/ f .x/ for all x Rn : (2.13) h Ni Ä N 2 e collection of all the subgradients of f at x is called the of the function at this N point and is denoted by @f.x/. N 54 2. SUBDIFFERENTIAL CALCULUS
Figure 2.7: Subdifferential of the absolute value function at the origin.
Note that it suffices to take only x dom f in the subgradient definition (2.13). Observe 2 also that @f.x/ under more general conditions; see, e.g., Proposition 2.47 as well as other N ¤ ; results in this direction given in [26]. e next proposition shows that the subdifferential @f.x/ can be equivalently defined ge- N ometrically via the normal cone to the epigraph of f at .x; f .x//. N N
Proposition 2.31 For any convex function f Rn R and any x dom f , we have W ! N 2 @f.x/ ˚v Rn ˇ .v; 1/ N .x; f .x// epi f «: (2.14) N D 2 ˇ 2 N N I
Proof. Fix v @f.x/ and .x; / epi f . Since f .x/, we deduce from (2.13) that 2 N 2 ˝.v; 1/; .x; / .x; f .x//˛ v; x x . f .x// N N D h Ni N v; x x .f .x/ f .x// 0: Ä h Ni N Ä is readily implies that .v; 1/ N..x; f .x// epi f /. 2 N N I To verify the opposite inclusion in (2.14), take an arbitrary element v Rn such that 2 .v; 1/ N..x; f .x// epi f /. For any x dom f , we have .x; f .x// epi f . us 2 N N I 2 2 v; x x .f .x/ f .x// ˝.v; 1/; .x; f .x// .x; f .x//˛ 0; h Ni N D N N Ä which shows that v @f.x/ and so justifies representation (2.14). 2 N Along with representing the subdifferential via the normal cone in (2.14), there is the fol- lowing simple way to express the normal cone to a set via the subdifferential (as well as the singular subdifferential (2.12)) of the extended-real-indicator function. 2.4. SUBGRADIENTS OF CONVEX FUNCTIONS 55 Example 2.32 Given a nonempty, convex set ˝ Rn, define its indicator function by (0 if x ˝; f .x/ ı.x ˝/ 2 (2.15) D I WD otherwise: 1 en epi f ˝ Œ0; /, and we have by Proposition 2.11(i) that D 1 N .x; f .x// epi f N.x ˝/ . ; 0; N N I D NI 1 which readily implies the relationships
@f.x/ @1f .x/ N.x ˝/: (2.16) N D N D NI A remarkable feature of convex functions is that every local minimizer for a convex function gives also the global/absolute minimum to the function.
Proposition 2.33 Let f Rn R convex and let x dom f be a local minimizer of f . en f W ! N 2 attains its global minimum at this point.
Proof. Since x is a local minimizer of f , there is ı > 0 such that N f .u/ f .x/ for all u IB.x ı/: N 2 NI n 1 1 Fix x R and construct a sequence of xk .1 k /x k x as k N. us we have xk 2 WD N C 2 2 IB.x ı/ when k is sufficiently large. It follows from the convexity of f that NI 1 1 f .x/ f .xk/ .1 k /f .x/ k f .x/; N Ä Ä N C which readily implies that k 1f .x/ k 1f .x/, and hence f .x/ f .x/ for all x Rn. N Ä N Ä 2 Next we recall the classical notion of derivative for functions on Rn, which we understand throughout the book in the sense of Fréchet unless otherwise stated; see Section 3.1.
Definition 2.34 We say that f Rn R is .Fréchet/ at x int.dom f / if there W ! N 2 exists an element v Rn such that 2 f .x/ f .x/ v; x x lim N h Ni 0: x x x x D !N k Nk In this case the element v is uniquely defined and is denoted by f .x/ v. r N WD e Fermat stationary rule of classical analysis tells us that if f Rn R is differentiable W ! at x and attains its local minimum at this point, then f .x/ 0. e following proposition N r N D 56 2. SUBDIFFERENTIAL CALCULUS gives a subdifferential counterpart of this result for general convex functions. Its proof is a direct consequence of the definitions and Proposition 2.33.
Proposition 2.35 Let f Rn R be convex and let x dom f . en f attains its local/global W ! N 2 minimum at x if and only if 0 @f.x/. N 2 N
Proof. Suppose that f attains its global minimum at x. en N f .x/ f .x/ for all x Rn; N Ä 2 which can be rewritten as 0 0; x x f .x/ f .x/ for all x Rn : D h Ni Ä N 2 e definition of the subdifferential shows that this is equivalent to 0 @f.x/. 2 N Now we show that the subdifferential (2.13) is indeed a singleton for differentiable func- tions reducing to the classical derivative/gradient at the reference point and clarifying the notion of differentiability in the case of convex functions.
Proposition 2.36 Let f Rn R be convex and differentiable at x int.dom f /. en we have W ! N 2 @f.x/ f .x/ and N D fr N g f .x/; x x f .x/ f .x/ for all x Rn : (2.17) hr N Ni Ä N 2 Proof. It follows from the differentiability of f at x that for any > 0 there is ı > 0 with N x x f .x/ f .x/ f .x/; x x x x whenever x x < ı: (2.18) k Nk Ä N hr N Ni Ä k Nk k Nk Consider further the convex function '.x/ f .x/ f .x/ f .x/; x x x x ; x Rn; WD N hr N Ni C k Nk 2 and observe that '.x/ '.x/ 0 for all x IB.x ı/. e convexity of ' ensures that '.x/ N D 2 NI '.x/ for all x Rn. us N 2 f .x/; x x f .x/ f .x/ x x whenever x Rn; hr N Ni Ä N C k Nk 2 which yields (2.17) by letting 0. # It follows from (2.17) that f .x/ @f.x/. Picking now v @f.x/, we get r N 2 N 2 N v; x x f .x/ f .x/: h Ni Ä N en the second part of (2.18) gives us that v f .x/; x x x x whenever x x < ı: h r N Ni Ä k Nk k Nk Finally, we observe that v f .x/ , which yields v f .x/ since > 0 was chosen ar- k r N k Ä D r N bitrarily. us @f.x/ f .x/ . N D fr N g 2.4. SUBGRADIENTS OF CONVEX FUNCTIONS 57 Corollary 2.37 Let f Rn R be a strictly convex function. en the differentiability of f at W ! x int.dom f / implies the strict inequality N 2 f .x/; x x < f .x/ f .x/ whenever x x: (2.19) hr N Ni N ¤ N
Proof. Since f is convex, we get from Proposition 2.36 that
f .x/; u x f .u/ f .x/ for all u Rn : hr N Ni Ä N 2 Fix x x and let u .x x/=2. It follows from the above that ¤ N WD CN x x x x 1 1 D f .x/; CN xE f CN Á f .x/ < f .x/ f .x/ f .x/: r N 2 N Ä 2 N 2 C 2 N N us we arrive at the strict inequality (2.19) and complete the proof. A simple while important example of a nonsmooth convex function is the norm function on Rn. Here is the calculation of its subdifferential.
Example 2.38 Let p.x/ x be the Euclidean norm function on Rn. en we have WD k k 8IB if x 0; @p.x/ < x D D n o otherwise: x : k k To verify this, observe first that the Euclidean norm function p is differentiable at any nonzero x point with p.x/ as x 0. It remains to calculate its subdifferential at x 0. To proceed r D x ¤ D by definition (2.13), wek k have that v @p.0/ if and only if 2 v; x v; x 0 p.x/ p.0/ x for all x Rn : h i D h i Ä D k k 2 Letting x v gives us v; v v , which implies that v 1, i.e., v IB. Now take v IB D h i Ä k k k k Ä 2 2 and deduce from the Cauchy-Schwarz inequality that
v; x 0 v; x v x x p.x/ p.0/ for all x Rn h i D h i Ä k k k k Ä k k D 2 and thus v @p.0/, which shows that @p.0/ IB. 2 D e next theorem calculates the subdifferential of the distance function.
eorem 2.39 Let ˝ be a nonempty, closed, convex subset of Rn. en we have
8N.x ˝/ IB if x ˝; @d.x ˝/ < x NI ˘.x\ ˝/ N 2 NI D n N NI o otherwise: : d.x ˝/ NI 58 2. SUBDIFFERENTIAL CALCULUS Proof. Consider the case where x ˝ and fix any element w @d.x ˝/. It follows from the N 2 2 NI definition of the subdifferential that w; x x d.x ˝/ d.x ˝/ d.x ˝/ for all x Rn : (2.20) h Ni Ä I NI D I 2 Since the distance function d. ˝/ is Lipschitz continuous with constant ` 1, we have I D w; x x x x for all x Rn : h Ni Ä k Nk 2 is implies that w 1, i.e., w IB. It also follows from (2.20) that k k Ä 2 w; x x 0 for all x ˝: h Ni Ä 2 us w N.x ˝/, which verifies that w N.x ˝/ IB. 2 NI 2 NI \ To prove the opposite inclusion, fix any w N.x ˝/ IB. en w 1 and 2 NI \ k k Ä w; u x 0 for all u ˝: h Ni Ä 2 us for every x Rn and every u ˝ we have 2 2 w; x x w; x u u x w; x u w; u x h Ni D h C Ni D h i C h Ni w; x u w x u x u : Ä h i Ä k k k k Ä k k Since u is taken arbitrarily in ˝, this implies w; x x d.x ˝/ d.x ˝/ d.x ˝/, and h Ni Ä I D I NI hence w @d.x ˝/. 2 NI Suppose now that x ˝, take z ˘.x ˝/ ˝, and fix any w @d.x ˝/. en N … N WD NI 2 2 NI w; x x d.x ˝/ d.x ˝/ d.x ˝/ x z h Ni Ä I NI D I k N Nk x z x z for all x Rn : Ä k Nk k N Nk 2 Denoting p.x/ x z , we have WD k Nk w; x x p.x/ p.x/ for all x Rn; h Ni Ä N 2 x z x z which ensures that w @p.x/ n N N o. Let us show that w N N is a subgradient of 2 N D x z D x z kn N Nk k N Nk d. ˝/ at x. Indeed, for any x R denote px ˘.x ˝/ and get I N 2 WD I w; x x w; x z w; z x w; x z x z h Ni D h Ni C h N Ni D h Ni k N Nk w; x px w; px z x z : D h i C h Ni k N Nk Since z ˘.x ˝/, it follows from Proposition 1.78 that x z; px z 0, and so we have N D NI h N N Ni Ä w; px z 0. Using the fact that w 1 and the Cauchy-Schwarz inequality gives us h Ni Ä k k D w; x x w; x px w; px z x z h Ni D h i C h Ni k N Nk w x px x z x px x z Ä k k k k k N Nk D k k k N Nk d.x ˝/ d.x ˝/ for all x Rn : D I NI 2 us we arrive at w @d.x ˝/ and complete the proof of the theorem. 2 NI 2.4. SUBGRADIENTS OF CONVEX FUNCTIONS 59 We conclude this section by providing first-order and second-order differential charac- terizations of convexity for differentiable and twice continuously differentiable .C 2/ functions, respectively, on given convex regions of Rn.
eorem 2.40 Let f Rn R be a differentiable function on its domain D, which is an open W ! convex set. en f is convex if and only if
f .u/; x u f .x/ f .u/ for all x; u D: (2.21) hr i Ä 2 Proof. e “only if ” part follows from Proposition 2.36. To justify the converse, suppose that (2.21) holds and then fix any x1; x2 D and t .0; 1/. Denoting xt tx1 .1 t/x2, we have 2 2 WD C xt D by the convexity of D. en 2
f .xt /; x1 xt f .x1/ f .xt /; f .xt /; x2 xt f .x2/ f .xt /: hr i Ä hr i Ä It follows furthermore that
t f .xt /; x1 xt tf .x1/ tf .xt / and hr i Ä .1 t/ f .xt /; x2 xt .1 t/f .x2/ .1 t/f .xt /: hr i Ä Summing up these inequalities, we arrive at
0 tf .x1/ .1 t/f .x2/ f .xt /; Ä C which ensures that f .xt / tf .x1/ .1 t/f .x2/, and so verifies the convexity of f . Ä C
Lemma 2.41 Let f Rn R be convex and twice continuously differentiable on an open subset V W ! of its domain containing x. en we have N 2f .x/u; u 0 for all u Rn; hr N i 2 where 2f .x/ is the Hessian matrix of f at x. r N N
Proof. Let A 2f .x/, which is symmetric matrix. en WD r N 1 f .x h/ f .x/ f .x/; h Ah; h lim N C N hr N i 2h i 0: (2.22) h 0 h 2 D ! k k It follows from (2.22) that for any > 0 there is ı > 0 such that 1 h 2 f .x h/ f .x/ f .x/; h Ah; h h 2 for all h ı: k k Ä N C N hr N i 2h i Ä k k k k Ä 60 2. SUBDIFFERENTIAL CALCULUS By Proposition 2.36, we readily have
f .x h/ f .x/ f .x/; h 0: N C N hr N i Combining the above inequalities ensures that 1 h 2 Ah; h whenever h ı: (2.23) k k Ä 2h i k k Ä u Observe further that for any 0 u Rn the element h ı satisfies h ı and, being ¤ 2 WD u k k Ä substituted into (2.23), gives us the estimate k k 1 u u ı2 ı2DA ; E: Ä 2 u u k k k k It shows therefore (since the case where u 0 is trivial) that D 2 u 2 Au; u whenever u Rn; k k Ä h i 2 which implies by letting 0 that 0 Au; u for all u Rn. # Ä h i 2 Now we are ready to justify the aforementioned second-order characterization of convexity for twice continuously differentiable functions.
eorem 2.42 Let f Rn R be a function twice continuously differentiable on its domain W ! D, which is an open convex subset of Rn. en f is convex if and only if 2f .x/ is positive r N semidefinite for every x D. N 2
Proof. Taking Lemma 2.41 into account, we only need to verify that if 2f .x/ is positive r N semidefinite for every x D, then f is convex. To proceed, for any x1; x2 D define xt N 2 2 WD tx1 .1 t/x2 and consider the function C '.t/ f tx1 .1 t/x2 tf .x1/ .1 t/f .x2/; t R : WD C 2
It is clear that ' is well defined on an open interval I containing .0; 1/. en '0.t/ 2 D f .xt /; x1 x2 f .x1/ f .x2/ and '00.t/ f .xt /.x1 x2/; x1 x2 0 for every t hr 2 i C D hr i 2 I since f .xt / is positive semidefinite. It follows from Corollary 1.46 that ' is convex on I . r Since '.0/ '.1/ 0, for any t .0; 1/ we have D D 2 '.t/ ' t.1/ .1 t/0 t'.1/ .1 t/'.0/ 0; D C Ä C D which implies in turn the inequality
f tx1 .1 t/x2 tf .x1/ .1 t/f .x2/; C Ä C is justifies that the function f is convex on its domain and thus on Rn. 2.5. BASIC CALCULUS RULES 61 2.5 BASIC CALCULUS RULES Here we derive basic calculus rules for the convex subdifferential (2.13) and its singular counter- part (2.12). First observe the obvious subdifferential rule for scalar multiplication.
Proposition 2.43 For a convex function f Rn R with x dom f , we have W ! N 2
@.˛f /.x/ ˛@f.x/ and @1.˛f /.x/ ˛@1f .x/ whenever ˛ > 0: N D N N D N Let us now establish the fundamental subdifferential sum rules for both constructions (2.13) and (2.12) that play a crucial role in convex analysis. We give a geometric proof for this result re- ducing it to the intersection rule for normals to sets, which in turn is based on convex separation of extremal set systems. Observe that our variational approach does not require any closedness/lower semicontinuity assumptions on the functions in question.
n eorem 2.44 Let fi R R for i 1; 2 be convex functions and let x dom f1 dom f2. W ! D N 2 \ Impose the singular subdifferential qualification condition
@1f1.x/ @1f2.x/ 0 : (2.24) N \ N D f g en we have the subdifferential sum rules
@.f1 f2/.x/ @f1.x/ @f2.x/; @1.f1 f2/.x/ @1f1.x/ @1f2.x/: (2.25) C N D N C N C N D N C N
Proof. For definiteness we verify the first rule in (2.25); the proof of the second result is similar. Fix an arbitrary subgradient v @.f1 f2/.x/ and define the sets 2 C N n ˝1 .x; 1; 2/ R R R 1 f1.x/ ; WD f 2 n j g ˝2 .x; 1; 2/ R R R 2 f2.x/ : WD f 2 j g Let us first verify the inclusion
.v; 1; 1/ N .x; f1.x/; f2.x// ˝1 ˝2: (2.26) 2 N N N I \
For any .x; 1; 2/ ˝1 ˝2, we have 1 f1.x/ and 2 f2.x/. en 2 \
v; x x . 1/.1 f1.x// . 1/.2 f2.x// h Ni C N C N v; x x f1.x/ f2.x/ f1.x/ f2.x/ 0; Ä h Ni C N N Ä where the last estimate holds due to v @.f1 f2/.x/. us (2.26) is satisfied. 2 C N To proceed with employing the intersection rule in (2.26), let us first check that the assumed qualification condition (2.24) yields
N .x; f1.x/; f2.x// ˝1 N .x; f1.x/; f2.x// ˝2 0 ; (2.27) N N N I \ N N N I D f g 62 2. SUBDIFFERENTIAL CALCULUS which is the basic qualification condition (2.9) needed for the application of Corollary 2.16 in the setting of (2.26). Indeed, we have from ˝1 epi f1 R that D
N .x; f1.x/; f2.x// ˝1 N .x; f1.x// epi f1 0 N N N I D N N I f g and observe similarly the representation
n N .x; f1.x/; f2.x// ˝2 ˚.v; 0; / R R R ˇ .v; / N .x; f2.x// epi f2«: N N N I D 2 ˇ 2 N N I Further, fix any element
.v; 1; 2/ N .x; f1.x/; f2.x// ˝1 N .x; f1.x/; f2.x// ˝2: 2 N N N I \ N N N I
en 1 2 0, and hence D D
.v; 0/ N .x; f1.x// epi f1 and . v; 0/ N .x; f2.x// epi f2: 2 N N I 2 N N I It follows by the definition of the singular subdifferential that
v @1f1.x/ @1f2.x/ 0 : 2 N \ N D f g n It yields .v; 1; 2/ .0; 0; 0/ R R R. Applying Corollary 2.16 in (2.26) shows that D 2
N .x; f1.x/; f2.x// ˝1 ˝2 N .x; f1.x/; f2.x// ˝1 N .x; f1.x/; f2.x// ˝2; N N N I \ D N N N I C N N N I which implies in turn that
.v; 1; 1/ .v1; 1; 0/ .v2; 0; 2/ D C
with .v1; 1/ N..x; f1.x// epi f1/ and .v2; 2/ N..x; f1.x// epi f2/. en we get 1 2 N N I 2 N N I D 2 1 and v v1 v2, where vi @fi .x/ for i 1; 2. is verifies the inclusion “ ” in the first D D C 2 N D subdifferential sum rule of (2.25). e opposite inclusion therein follows easily from the definition of the subdifferential. In- deed, any v @f1.x/ @f2.x/ can be represented as v v1 v2, where vi @fi .x/ for i 1; 2. 2 N C N D C 2 N D en we have
v; x x v1; x x v2; x x h Ni D h Ni C h Ni f1.x/ f1.x/ f2.x/ f2.x/ .f1 f1/.x/ .f1 f2/.x/ Ä N C N D C C N n for all x R , which readily implies that v @.f1 f2/.x/ and therefore completes the proof 2 2 C N of the theorem. Note that by Proposition 2.23 the qualification condition (2.24) can be written as
N.x dom f1/ N.x dom f2/ 0 : NI \ NI D f g 2.5. BASIC CALCULUS RULES 63 Let us present some useful consequences of eorem 2.44. e sum rule for the (basic) subdif- ferential (2.13) in the following corollary is known as the Moreau-Rockafellar theorem.
n Corollary 2.45 Let fi R R for i 1; 2 be convex functions such that there exists u W ! D 2 dom f1 dom f2 for which f1 is continuous at u or f2 is continuous at u. en \
@.f1 f2/.x/ @f1.x/ @f2.x/ (2.28) C D C n whenever x dom f1 dom f2. Consequently, if both functions fi are finite-valued on R , then the 2 \ sum rule (2.28) holds for all x Rn. 2
Proof. Suppose for definiteness that f1 is continuous at u dom f1 dom f2, which implies 2 \ that u int.dom f1/ dom f2. en the proof of Corollary 2.18 shows that 2 \
N.x dom f1/ N.x dom f2/ 0 whenever x dom f1 dom f2: I \ I D f g 2 \ us the conclusion follows directly from eorem 2.44. e next corollary for sums of finitely many functions can be derived from eorem 2.44 by induction, where the validity of the qualification condition in the next step of induction follows from the singular subdifferential sum rule in (2.25).
n Corollary 2.46 Consider finitely many convex functions fi R R for i 1; : : : ; m. e fol- W ! D lowing assertions hold: m (i) Let x Ti 1 dom fi and let the implication N 2 D m h X vi 0; vi @1fi .x/i vi 0 for all i 1; : : : ; m (2.29) D 2 N H) D D i 1 D hold. en we have the subdifferential sum rules
m m m m
@ X fi Á.x/ X @fi .x/; @1 X fi Á.x/ X @fi .x/: N D N N D N i 1 i 1 i 1 i 1 D D D D m (ii) Suppose that there is u T dom fi such that all (except possibly one) functions fi are continuous 2 i 1 at u. en we have the equalityD
m m
@ X fi Á.x/ X @fi .x/ D i 1 i 1 D D m for any common point of the domains x Ti 1 dom fi . 2 D 64 2. SUBDIFFERENTIAL CALCULUS e next result not only ensures the subdifferentiability of convex functions, but also justifies the compactness of its subdifferential at interior points of the domain.
Proposition 2.47 For any convex function f Rn R and any x int.dom f /, we have that W ! N 2 the subgradient set @f.x/ is nonempty and compact in Rn. N
Proof. Let us first verify the boundedness of the subdifferential. It follows from Corollary 2.27 that f is locally Lipschitz continuous around x with some constant `, i.e., there exists a positive N number such that
f .x/ f .y/ ` x y for all x; y IB.x /: j j Ä k k 2 NI us for any subgradient v @f.x/ and any element h Rn with h ı we have 2 N 2 k k Ä v; h f .x h/ f .x/ ` h : h i Ä N C N Ä k k is implies the bound v `. e closedness of @f.x/ follows directly from the definition, and k k Ä N so we get the compactness of the subgradient set. It remains to verify that @f.x/ . Since .x; f .x// bd .epi f /, we deduce from N ¤ ; N N 2 Corollary 2.14 that N..x; f .x// epi f / 0 . Take such an element .0; 0/ .v; / N N I ¤ f g ¤ 2 N..x; f .x// epi f / and get by the definition that N N I v; x x . f .x// 0 for all .x; / epi f: h Ni N Ä 2 Using this inequality with x x and f .x/ 1 yields 0. If 0, then v @ f .x/ WD N D N C D 2 1 N D 0 by eorem 2.29, which is a contradiction. us > 0 and .v= ; 1/ N..x; f .x// epi f /. f g 2 N N I Hence v= @f.x/, which completes the proof of the proposition. 2 N Given f Rn R and any ˛ R, define the level set W ! 2 n ˝˛ ˚x R ˇ f .x/ ˛«: WD 2 ˇ Ä
n Proposition 2.48 Let f R R be convex and let x ˝˛ with f .x/ ˛ for some ˛ R. If W ! N 2 N D 2 0 @f.x/, then the normal cone to the level set ˝˛ at x is calculated by … N N
N.x ˝˛/ @1f .x/ R> @f.x/; (2.30) NI D N [ N
where R> denotes the set of all positive real numbers. 2.5. BASIC CALCULUS RULES 65 n n 1 Proof. To proceed, consider the set R ˛ R C and observe that WD f g
˝˛ ˛ epi f : f g D \ Let us check the relationship
N .x; ˛/ epi f N .x; ˛/ 0 ; (2.31) N I \ N I D f g i.e., the qualification condition (2.9) holds for the sets epi f and . Indeed, take .v; / from the set on left-hand side of (2.31) and deduce from the structure of that v 0 and that 0. D If we suppose that > 0, then
1 .v; / .0; 1/ N .x; ˛/ epi f ; D 2 N I and so 0 @f.x/, which contradicts the assumption of this proposition. us we have .v; / 2 N D .0; 0/, and then applying the intersection rule of Corollary 2.16 gives us
N.x; ˝˛/ R N .x; ˛/ ˝˛ ˛ N .x; ˛/ epi f . 0 R/: N D N I f g D N I C f g
Fix now any v N.x ˝˛/ and get .v; 0/ N.x; ˝˛/ R. is implies the existence of 0 2 NI 2 N such that .v; / N..x; ˛/ epi f / and .v; 0/ .v; / .0; /. If 0, then v @ f .x/. 2 N I D C D 2 1 N In the case when > 0 we obtain v @f.x/ and thus verify the inclusion “ ” in (2.30). 2 N Let us finally prove the opposite inclusion in (2.30). Since ˝˛ dom f , we get
@1f .x/ N.x dom f / N.x ˝˛/: N D NI NI
Take any v R> @f.x/ and find > 0 such that v @f.x/. Taking any x ˝˛ gives us f .x/ 2 N 2 N 2 Ä ˛ f .x/. erefore, we have D N 1 v; x x f .x/ f .x/ 0; h Ni Ä N Ä which implies that v N.x ˝˛/ and thus completes the proof. 2 NI
Corollary 2.49 In the setting of Proposition 2.48 suppose that f is continuous at x and that 0 N … @f.x/. en we have the representation N
N.x ˝˛/ R @f.x/ [ ˛@f.x/; ˛ R : NI D C N D N 2 ˛ 0 Proof. Since f is continuous at x, it follows from eorem 2.29 that @ f .x/ 0 . en the N 1 N D f g result of this corollary follows directly from Proposition 2.48. 66 2. SUBDIFFERENTIAL CALCULUS We now obtain some other calculus rules for subdifferentiation of convex function. For brevity, let us only give results for the basic subdifferential (2.13) leaving their singular subdiffer- ential counterparts as exercises; see Section 2.11. To proceed with deriving chain rules, we present first the following geometric lemma.
Lemma 2.50 Let B Rn Rp be an affine mapping given by B.x/ A.x/ b, where A W ! D C W Rn Rp is a linear mapping and b Rp. For any .x; y/ gph B, we have ! 2 N N 2 n p N .x; y/ gph B ˚.u; v/ R R ˇ u Av«: N N I D 2 ˇ D
Proof. It is clear that the set gph B is convex and .u; v/ N..x; y/ gph B/ if and only if 2 N N u; x x v; B.x/ B.x/ 0 whenever x Rn : (2.32) h Ni C h N i Ä 2 It follows directly from the definitions that
u; x x v; B.x/ B.x/ u; x x v; A.x/ A.x/ h Ni C h N i D h Ni C h N i u; x x Av; x x D h Ni C h Ni u Av; x x ; D h C Ni which implies the equivalence of (2.32) to u A v; x x 0, and so to u A v. h C Ni Ä D
eorem 2.51 Let f Rp R be a convex function and let B Rn Rp be as in Lemma 2.50 W ! W ! with B.x/ dom f for some x Rp. Denote y B.x/ and assume that N 2 N 2 N WD N
ker A @1f .y/ 0 : (2.33) \ N D f g en we have the subdifferential chain rule
@.f B/.x/ A @f.y/ ˚Av ˇ v @f.y/«: (2.34) ı N D N D ˇ 2 N
Proof. Fix v @.f B/.x/ and form the subsets of Rn Rp R by 2 ı N n ˝1 gph B R and ˝2 R epi f: WD WD It follows from the definition of the subdifferential and the normal cone that .v; 0; 1/ 2 N..x; y; z/ ˝1 ˝2/, where z f .y/. Indeed, fix any .x; y; / ˝1 ˝2. en y B.x/ N N N I \ N WD N 2 \ D and f .y/, and so f .B.x//. us v; x x 0.y y/ . 1/. z/ v; x x Œf .B.x// f .B.x// 0: h Ni C N C N Ä h Ni N Ä 2.5. BASIC CALCULUS RULES 67 Employing the intersection rule from Corollary 2.19 under (2.33) gives us
.v; 0; 1/ N .x; y; z/ ˝1 N .x; y; z/ ˝2; 2 N N N I C N N N I which reads that .v; 0; 1/ .v; w; 0/ .0; w; 1/ with .v; w/ N..x; y/ gph B/ and D C 2 N N I .w; 1/ N..y; z/ epi f /. en we get 2 N N I
v Aw and w @f.y/; D 2 N which implies in turn that v A [email protected]// and thus verifies the inclusion “ ” in (2.34). e op- 2 N posite inclusion follows directly from the definition of the subdifferential. In the corollary below we present some conditions that guarantee that the qualification (2.33) in eorem 2.51 is satisfied.
Corollary 2.52 In the setting of eorem 2.51 suppose that either A is surjective or f is continuous at the point y Rn; the latter assumption is automatic when f is finite-valued. en we have the N 2 subdifferential chain rule (2.34).
Proof. Let us check that the qualification condition (2.33) holds under the assumptions of this corollary. e surjectivity of A gives us ker A 0 , and in the second case we have D f g y int.dom f /, so @ f .y/ 0 . e validity of (2.33) is obvious. N 2 1 N D f g e last consequence of eorem 2.51 presents a chain rule for normals to sets.
Corollary 2.53 Let ˝ Rp be a convex set and let B Rn Rp be an affine mapping as in W ! Lemma 2.50 satisfying y B.x/ ˝. en we have N WD N 2 1 N.x B .˝// A.N.y ˝// NI D NI under the validity of the qualification condition
ker A N.y ˝/ 0 : \ NI D f g
Proof. is follows from eorem 2.51 with f .x/ ı.x ˝/. D I n Given fi R R for i 1; : : : ; m, consider the maximum function W ! D n f .x/ max fi .x/; x R ; (2.35) WD i 1;:::;m 2 D and for x Rn define the set N 2
I.x/ ˚i 1; : : : ; m ˇ fi .x/ f .x/«: N WD 2 f g ˇ N D N 68 2. SUBDIFFERENTIAL CALCULUS
In the proposition below we assume that each function fi for i 1; : : : ; m is continuous D at the reference point. A more general version follows afterward.
n Proposition 2.54 Let fi R R, i 1; : : : ; m, be convex functions. Take any point x m W ! D N 2 Ti 1 dom fi and assume that each fi is continuous at x. en we have the maximum rule D N
@ max fi /.x/ co [ @fi .x/: N D N i I.x/ 2 N Proof. Let f .x/ be the maximum function defined in (2.35). We obviously have
m
epi f \ epi fi : D i 1 D Since fi .x/ < f .x/ ˛ for any i I.x/, there exist a neighborhood U of x and ı > 0 such N N DW N … N N that fi .x/ < ˛ whenever .x; ˛/ U .˛ ı; ˛ ı/. It follows that .x; ˛/ int epi fi , and so 2 N N C N N 2 N..x; ˛/ epi fi / .0; 0/ for such indices i. N N I D f g Let us show that the qualification condition from Corollary 2.19 is satisfied for the sets ˝i epi fi , i 1; : : : ; m, at .x; ˛/. Fix .vi ; i / N .x; ˛/ epi fi with WD D N N 2 N N I m
X.vi ; i / 0: D i 1 D en .vi ; i / .0; 0/ for i I.x/, and hence D … N
X .vi ; i / 0: D i I.x/ 2 N
Note that fi .x/ f .x/ ˛ for i I.x/. e proof of Proposition 2.47 tells us that i 0 N D N DN 2 N for i I.x/. us the equality Pi I.x/ i 0 yields i 0, and so vi @1fi .x/ 0 for 2 N 2 N D D 2 N D f g every i I.x/ by eorem 2.29. is verifies that .vi ; i / .0; 0/ for i 1; : : : ; m. It follows 2 N D D furthermore that m N .x; f .x// epi f X N .x; ˛/ epi fi X N .x; fi .x// epi fi : N N I D N N I D N N I i 1 i I.x/ D 2 N
For any v @f.x/, we have .v; 1/ N..x; f .x// epi f /, which allows us to find .vi ; i / 2 N 2 N N I 2 N..x; fi .x// epi fi / for i I.x/ such that N N I 2 N
.v; 1/ X .vi ; i /: D i I.x/ 2 N
is yields Pi I.x/ i 1 and v Pi I.x/ vi . If i 0, then vi @1fi .x/ 0 . Hence we 2 N D D 2 N D 2 N D f g assume without loss of generality that i > 0 for every i I.x/. Since N..x; fi .x// epi fi / is a 2 N N N I 2.5. BASIC CALCULUS RULES 69 cone, it gives us .vi =i ; 1/ N..x; fi .x// epi fi /, and so vi i @fi .x/. We get the represen- 2 N N I 2 N tation v Pi I.x/ i ui , where ui @fi .x/ and Pi I.x/ i 1. us D 2 N 2 N 2 N D
v co [ @fi .x/: 2 N i I.x/ 2 N Note that the opposite inclusion follows directly from
@fi .x/ @f.x/ for all i I.x/; N N 2 N which in turn follows from the definition. Indeed, for any i I.x/ and v @fi .x/ we have 2 N 2 N n v; x x fi .x/ fi .x/ fi .x/ f .x/ f .x/ f .x/ whenever x R ; h Ni Ä N D N Ä N 2 which implies that v @f.x/ and thus completes the proof. 2 N To proceed with the general case, define
m .x/ ˚.1; : : : ; m/ ˇ i 0; X i 1; i fi .x/ f .x/ 0« N WD ˇ D N N D i 1 D and observe from the definition of .x/ hat N .x/ ˚.1; : : : ; m/ ˇ i 0; X i 1; i 0 for i I.x/«: N D ˇ D D … N i I.x/ n 2 N eorem 2.55 Let fi R R be convex functions for i 1; : : : ; m. Take any point x m W ! D N 2 T dom fi and assume that each fi is continuous at x for any i I.x/ and that the quali- i 1 N … N ficationD condition
h X vi 0; vi @1fi .x/i vi 0 for all i I.x/ D 2 N H) D 2 N i I.x/ 2 N holds. en we have the maximum rule
@ max fi .x/ [ n X i @fi .x/ ˇ .1; : : : ; m/ .x/o; N D ı N ˇ 2 N i I.x/ ˇ 2 N where i @fi .x/ i @fi .x/ if i > 0, and i @fi .x/ @ fi .x/ if i 0. ı N WD N ı N WD 1 N D
Proof. Let f .x/ be the maximum function from (2.35). Repeating the first part of the proof of Proposition 2.54 tells us that N..x; f .x// epi fi / .0; 0/ for i I.x/. e imposed qualifica- N N I D f g … N tion condition allows us to employ the intersection rule from Corollary 2.19 to get
m N .x; f .x// epi f X N .x; f .x// epi fi X N .x; fi .x// epi fi : (2.36) N N I D N N I D N N I i 1 i I.x/ D 2 N 70 2. SUBDIFFERENTIAL CALCULUS For any v @f.x/, we have .v; 1/ N..x; f .x// epi f /, and thus find by using (2.36) such 2 N 2 N N I pairs .vi ; i / N..x; fi .x// epi fi / for i I.x/ that 2 N N I 2 N
.v; 1/ X .vi ; i /: D i I.x/ 2 N
is shows that P i 1 and v P vi with vi i @fi .x/ and i 0 for all i i I.x/ D D i I.x/ 2 ı N 2 I.x/. erefore, we2 arriveN at 2 N N
v [ n X i @fi .x/ ˇ .1; : : : ; m/ .x/o; 2 ı N ˇ 2 N i I.x/ ˇ 2 N which justifies the inclusion “ ” in the maximum rule. e opposite inclusion can also be verified easily by using representations (2.36).
2.6 SUBGRADIENTS OF OPTIMAL VALUE FUNCTIONS is section is mainly devoted to the study of the class of optimal value/marginal functions defined in (1.4). To proceed, we first introduce and examine the notion of coderivatives for set-valued mappings, which is borrowed from variational analysis and plays a crucial role in many aspects of the theory and applications of set-valued mappings; in particular, those related to subdifferenti- ation of optimal value functions considered below.
Definition 2.56 Let F Rn Rp be a set-valued mapping with convex graph and let .x; y/ W ! N N 2 gph F . e of F!at .x; y/ is the set-valued mapping D F.x; y/ Rp Rn with the N N N N W ! values ! n p DF.x; y/.v/ ˚u R ˇ .u; v/ N .x; y/ gph F «; v R : (2.37) N N WD 2 ˇ 2 N N I 2 It follows from (2.37) and definition (2.4) of the normal cone to convex sets that the coderivative values of convex-graph mappings can be calculated by
n DF.x; y/.v/ nu R ˇ u; x v; y max u; x v; y o: N N D 2 ˇ h Ni h Ni D .x;y/ gph F h i h i ˇ 2 Now we present explicit calculations of the coderivative in some particular settings.
Proposition 2.57 Let F Rn Rp be a single-valued mapping given by F.x/ Ax b, where W ! WD C A Rn Rp is a linear mapping, and where b Rp. For any pair .x; y/ such that y Ax b, W ! 2 N N N D N C we have the representation
p DF.x; y/.v/ Av whenever v R : N N D f g 2 2.6. SUBGRADIENTS OF OPTIMAL VALUE FUNCTIONS 71 Proof. Lemma 2.50 tells us that
n p N .x; y/ gph F/ ˚.u; v/ R R ˇ u Av«: N N I D 2 ˇ D us u D F.x; y/.v/ if and only if u A v. 2 N N D
Proposition 2.58 Given a convex function f Rn R with x dom f , define the set-valued W ! N 2 mapping F Rn R by W ! F.x/ f .x/; : WD 1 Denoting y f .x/, we have N WD N 8@f.x/ if > 0; ˆ N DF.x; y/./ <@ f .x/ if 0; N N D 1 N D ˆ if < 0: :; In particular, if f is finite-valued, then D F.x; y/./ @f.x/ for all 0. N N D N
Proof. It follows from the definition of F that
gph F ˚.x; / Rn R ˇ F.x/« ˚.x; / Rn R ˇ f .x/« epi f: D 2 ˇ 2 D 2 ˇ D us the inclusion v D F.x; y/./ is equivalent to .v; / N..x; y/ epi f /. If > 0, then 2 N N 2 N N I .v=; 1/ N..x; y/ epi f /, which implies v= @f.x/, and hence v @f.x/. e formula 2 N N I 2 N 2 N in the case where 0 follows from the definition of the singular subdifferential. If < 0, then D D F.x; y/./ because 0 whenever .v; / N..x; y/ epi f / by the proof of Propo- N N D; 2 N N I sition 2.47. To verify the last part of the proposition, observe that in this case D F.x; y/.0/ N N D @ f .x/ 0 by Proposition 2.23, so we arrive at the conclusion. 1 N D f g e next simple example is very natural and useful in what follows.
Example 2.59 Define F Rn Rp by F .x/ K for every x Rn, where K Rp W ! D 2 is a nonempty, convex set. en! gph F Rn K, and Proposition 2.11(i) tells us that D N..x; y/ gph F/ 0 N.y K/ for any .x; y/ Rn K. us N N I D f g NI N N 2 ( 0 if v N.y K/; DF.x; y/.v/ f g 2 NI N N D otherwise: ; In particular, for the case where K Rp we have D ( 0 if v 0; DF.x; y/.v/ f g D N N D otherwise: ; 72 2. SUBDIFFERENTIAL CALCULUS
Now we derive a formula for calculating the subdifferential of the optimal value/marginal function (1.4) given in the form .x/ inf ˚'.x; y/ ˇ y F.x/« WD ˇ 2 via the coderivative of the set-valued mapping F and the subdifferential of the function '. First we derive the following useful estimate.
Lemma 2.60 Let . / be the optimal value function (1.4) generated by a convex-graph mapping F Rn Rp and a convex function ' Rn Rp R. Suppose that .x/ > for all x Rn, W ! W ! 1 2 fix some x! dom , and consider the solution set N 2 S.x/ ˚y F.x/ ˇ .x/ '.x; y/«: N WD N 2 N ˇ N D N N If S.x/ , then for any y S.x/ we have N ¤ ; N 2 N [ u DF.x; y/.v/ @.x/: (2.38) C N N N .u;v/ @'.x;y/ 2 N N Proof. Pick w from the left-hand side of (2.38) and find .u; v/ @'.x; y/ with 2 N N w u DF.x; y/.v/: 2 N N It gives us .w u; v/ N..x; y/ gph F/ and thus 2 N N I w u; x x v; y y 0 for all .x; y/ gph F; h Ni h Ni Ä 2 which shows that whenever y F .x/ we have 2 w; x x u; x x v; y y '.x; y/ '.x; y/ '.x; y/ .x/: h Ni Ä h Ni C h Ni Ä N N D N is implies in turn the estimate w; x x inf '.x; y/ .x/ .x/ .x/; h Ni Ä y F .x/ N D N 2 which justifies the inclusion w @.x/, and hence completes the proof of (2.38). 2 N eorem 2.61 Consider the optimal value function (1.4) under the assumptions of Lemma 2.60. For any y S.x/, we have the equality N 2 N @.x/ [ u DF.x; y/.v/ (2.39) N D C N N .u;v/ @'.x;y/ 2 N N provided the validity of the qualification condition
@1'.x; y/ N..x; y/ gph F/ .0; 0/ ; (2.40) N N \ N N I D f g which holds, in particular, when ' is continuous at .x; y/. N N 2.6. SUBGRADIENTS OF OPTIMAL VALUE FUNCTIONS 73 Proof. Taking Lemma 2.60 into account, it is only required to verify the inclusion “ ” in (2.39). To proceed, pick w @.x/ and y S.x/. For any x Rn, we have 2 N N 2 N 2 w; x x .x/ .x/ .x/ '.x; y/ h Ni Ä N D N N '.x; y/ '.x; y/ for all y F .x/: Ä N N 2 is implies that, whenever .x; y/ Rn Rp, the following inequality holds: 2 w; x x 0; y y '.x; y/ ı .x; y/ gph F/ '.x; y/ ı .x; y/ gph F : h Ni C h Ni Ä C I N N C N N I Denote further f .x; y/ '.x; y/ ı..x; y/ gph F/ and deduce from the subdifferential sum WD C I rule in eorem 2.44 under the qualification condition (2.40) that the inclusion
.w; 0/ @f.x; y/ @'.x; y/ N .x; y/ gph F 2 N N D N N C N N I is satisfied. is shows that .w; 0/ .u1; v1/ .u2; v2/ with .u1; v1/ @'.x; y/ and .u2; v2/ D C 2 N N 2 N..x; y/ gph F/. is yields v2 v1, so .u2; v1/ N..x; y/ gph F/. Finally, we get u2 N N I D 2 N N I 2 D F.x; y/.v1/ and therefore N N
w u1 u2 u1 DF.x; y/.v1/; D C 2 C N N which completes the proof of the theorem. Let us present several important consequences of eorem 2.61. e first one concerns the following chain rule for convex compositions.
Corollary 2.62 Let f Rn R be a convex function and let ' R R be a nondecreasing convex W ! W ! function. Take x Rn and denote y f .x/ assuming that y dom '. en N 2 N WD N N 2 @.' f /.x/ [ @f.x/ ı N D N @'.y/ 2 N under the assumption that either @ '.y/ 0 or 0 @f.x/. 1 N D f g … N
Proof. e composition ' f is a convex function by Proposition 1.39. Define the set-valued ı mapping F.x/ Œf .x/; / and observe that WD 1 .' f /.x/ inf '.y/ ı D y F .x/ 2 since ' is nondecreasing. Note each of the assumptions @ '.y/ 0 and 0 @f.x/ ensure the 1 N D f g … N validity of the qualification condition (2.40). It follows from eorem 2.61 that
@.' f /.x/ [ DF.x; y/./: ı N D N N @'.y/ 2 N 74 2. SUBDIFFERENTIAL CALCULUS Taking again into account that ' is nondecreasing yields 0 for every @'.y/. is tells us 2 N by Proposition 2.58 that
@.' f /.x/ [ DF.x; y/./ [ @f.x/; ı N D N N D N @'.y/ @'.y/ 2 N 2 N which verifies the claimed chain rule of this corollary. e next corollary addresses calculating subgradients of the optimal value functions over parameter-independent constraints.
Corollary 2.63 Let ' Rn Rp R, where the function '.x; / is bounded below on Rp for every W ! x Rn. Define 2 .x/ inf ˚'.x; y/ ˇ y Rp «; S.x/ ˚y Rp ˇ '.x; y/ .x/«: WD ˇ 2 WD 2 ˇ D For x dom and for every y S.x/, we have N 2 N 2 N @.x/ ˚u Rn ˇ .u; 0/ @'.x; y/«: N D 2 ˇ 2 N N
Proof. Follows directly from eorem 2.61 for F .x/ Rp and Example 2.59. Note that the Á qualification condition (2.40) is satisfied since N..x; y/ gph F/ .0; 0/ . N N I D f g e next corollary specifies the general result for problems with affine constraints.
Corollary 2.64 Let B Rp Rn be given by B.y/ Ay b via a linear mapping A Rp Rn W ! WD C W ! and an element b Rn and let ' Rp R be a convex function bounded below on the inverse image 2 W ! set B 1.x/ for all x Rn. Define 2 1 .x/ inf ˚'.y/ ˇ B.y/ x ; S.x/ ˚y B .x/ ˇ '.y/ .x/« WD ˇ D g WD 2 ˇ D and fix x Rn with .x/ < and S.x/ . For any y S.x/, we have N 2 N 1 N ¤ ; N 2 N 1 @.x/ .A/ @'.y/: N D N
Proof. e setting of this corollary corresponds to eorem 2.61 with F .x/ B 1.x/ and D '.x; y/ '.y/. en we have Á n p N .x; y/ gph F ˚.u; v/ R R ˇ Au v«; N N I D 2 ˇ D which therefore implies the coderivative representation
n 1 p DF.x; y/.v/ ˚u R ˇ Au v« .A/ .v/; v R : N N D 2 ˇ D D 2 2.7. SUBGRADIENTS OF SUPPORT FUNCTIONS 75 It is easy to see that the qualification condition (2.40) is automatic in this case. is yields
1 @.x/ [ DF.x; y/.v/ .A/ .@'.y//; N D N N D N v @'.y/ 2 N and thus we complete the proof. Finally in this section, we present a useful application of Corollary 2.64 to deriving a cal- culus rule for subdifferentiation of infimal convolutions important in convex analysis.
n Corollary 2.65 Given convex functions f1; f2 R R, consider their infimal convolution W ! n .f1 f2/.x/ inf ˚f1.x1/ f2.x2/ ˇ x1 x2 x«; x R ; ˚ WD C ˇ C D 2 n n and suppose that .f1 f2/.x/ > for all x R . Fix x dom .f1 f2/ and x1; x2 R sat- ˚ 1 2 N 2 ˚ N N 2 isfying x x1 x2 and .f1 f2/.x/ f1.x1/ f2.x2/. en we have N DN CN ˚ N D N C N
@.f1 f2/.x/ @f1.x1/ @f2.x2/: ˚ N D N \ N
Proof. Define ' Rn Rn R and A Rn Rn Rn by W ! W !
'.x1; x2/ f1.x1/ f2.x2/; A.x1; x2/ x1 x2: WD C WD C n It is easy to see that Av .v; v/ for any v R and that @'.x1; x2/ [email protected]/; @f2.x2// for n n D 2 N N D N N .x1; x2/ R R . en v @.f1 f2/.x/ if and only if we have A v .v; v/ @'.x1; x2/, N N 2 2 ˚ N D 2 N N which means that v @f1.x1/ @f2.x2/. us the claimed calculus result follows from Corol- 2 N \ N lary 2.64.
2.7 SUBGRADIENTS OF SUPPORT FUNCTIONS is short section is devoted to the study of subgradient properties for an important class of nonsmooth convex functions known as support functions of convex sets.
Definition 2.66 Given a nonempty .not necessarily convex/ subset ˝ of Rn, the of ˝ is defined by ˝ .x/ sup ˚ x; ! ˇ ! ˝«: (2.41) WD h i ˇ 2 First we present some properties of (2.41) that can be deduced from the definition.
Proposition 2.67 e following assertions hold: n (i) For any nonempty subset ˝ of R , the support function ˝ is subadditive, positively homogeneous, and hence convex. 76 2. SUBDIFFERENTIAL CALCULUS n n (ii) For any nonempty subsets ˝1; ˝2 R and any x R , we have 2
˝1 ˝2 .x/ ˝1 .x/ ˝2 .x/; ˝1 ˝2 .x/ max ˚˝1 .x/; ˝2 .x/«: C D C [ D e next result calculates subgradients of support functions via convex separation.
eorem 2.68 Let ˝ be a nonempty, closed, convex subset of Rn and let
S.x/ ˚p ˝ ˇ x; p ˝ .x/«: WD 2 ˇ h i D
For any x dom ˝ , we have the subdifferential formula N 2
@˝ .x/ S.x/: N D N
Proof. Fix p @˝ .x/ and get from definition (2.13) that 2 N n p; x x ˝ .x/ ˝ .x/ whenever x R : (2.42) h Ni Ä N 2 Taking this into account, it follows for any u Rn that 2
p; u p; u x x ˝ .u x/ ˝ .x/ ˝ .u/ ˝ .x/ ˝ .x/ ˝ .u/: (2.43) h i D h CN Ni Ä CN N Ä C N N D Assuming that p ˝, we use the separation result of Proposition 2.1 and find u Rn with 62 2
˝ .u/ sup ˚ u; ! ˇ ! ˝« < p; u ; D h i ˇ 2 h i
which contradicts (2.43) and so verifies p ˝. From (2.43) we also get p; x ˝ .x/. Letting 2 h Ni Ä N x 0 in (2.42) gives us ˝ .x/ p; x . Hence p; x ˝ .x/ and thus p S.x/. D N Ä h Ni h Ni D N 2 N Now fix p S.x/ and get from p ˝ and p; x ˝ .x/ that 2 N 2 h Ni D N n p; x x p; x p; x ˝ .x/ ˝ .x/ for all x R : h Ni D h i h Ni Ä N 2
is shows that p @˝ .x/ and thus completes the proof. 2 N Note that eorem 2.68 immediately implies that
@˝ .0/ S.0/ ˚! ˝ ˇ 0; ! ˝ .0/ 0« ˝ (2.44) D D 2 ˇ h i D D D for any nonempty, closed, convex set ˝ Rn. e last result of this section justifies a certain monotonicity property of support functions with respect to set inclusions.
n Proposition 2.69 Let ˝1 and ˝2 be a nonempty, closed, convex subsets of R . en the inclusion n ˝1 ˝2 holds if and only if ˝ .v/ ˝ .v/ for all v R . 1 Ä 2 2 2.8. FENCHEL CONJUGATES 77 n Proof. Taking ˝1 ˝2 as in the proposition, for any v R we have 2 ˝ .v/ sup ˚ v; x ˇ x ˝1« sup ˚ v; x ˇ x ˝2« ˝ .v/: 1 D h i ˇ 2 Ä h i ˇ 2 D 2 n Conversely, suppose that ˝ .v/ ˝ .v/ whenever v R . Since ˝ .0/ ˝ .0/ 0, it fol- 1 Ä 2 2 1 D 2 D lows from definition (2.13) of the subdifferential that
@˝ .0/ @˝ .0/; 1 2 which yields ˝1 ˝2 by formula (2.44). 2.8 FENCHEL CONJUGATES Many important issues of convex analysis and its applications (in particular, to optimization) are based on duality. e following notion plays a crucial role in duality considerations.
Definition 2.70 Given a function f Rn R .not necessarily convex/, its F W ! f Rn Œ ; is W ! 1 1 n f .v/ sup ˚ v; x f .x/ ˇ x R « sup ˚ v; x f .x/ ˇ x dom f «: (2.45) WD h i ˇ 2 D h i ˇ 2 Note that f .v/ is allowed in (2.45) while f .v/ > for all v Rn if dom f D 1 1 2 ¤ . It follows directly from the definitions that for any nonempty set ˝ Rn the conjugate to its ; indicator function is the support function of ˝:
ı .v/ sup ˚ v; x ˇ x ˝« ˝ .v/; v ˝: (2.46) ˝ D h i ˇ 2 D 2 e next two propositions can be easily verified.
Proposition 2.71 Let f Rn R be a function, not necessarily convex, with dom f . en W ! n ¤ ; its Fenchel conjugate f is convex on R . Proof. Function (2.45) is convex as the supremum of a family of affine functions.
Proposition 2.72 Let f; g Rn R be such that f .x/ g.x/ for all x Rn. en we have W ! Ä 2 f .v/ g .v/ for all v Rn. 2
Proof. For any fixed v Rn, it follows from (2.45) that 2 v; x f .x/ v; x g.x/; x Rn : h i h i 2 is readily implies the relationships
n n f .v/ sup ˚ v; x f .x/ ˇ x R « sup ˚ v; x g.x/ ˇ x R « g.v/ D h i ˇ 2 h i ˇ 2 D for all v Rn, and therefore f g on Rn. 2 78 2. SUBDIFFERENTIAL CALCULUS
Figure 2.8: Fenchel conjugate.
e following two examples illustrate the calculation of conjugate functions.
Example 2.73 (i) Given a Rn and b R, consider the affine function 2 2 f .x/ a; x b; x Rn : WD h i C 2 en it can be seen directly from the definition that
( b if v a; f .v/ D D otherwise: 1 (ii) Given any p > 1, consider the power function xp 8 if x 0; f .x/ < p WD otherwise: :1 For any v R, the conjugate of this function is given by 2 xp xp n ˇ o n ˇ o f .v/ sup vx ˇ x 0 inf vx ˇ x 0 : D p ˇ D p ˇ 1 p It is clear that f .v/ 0 if v 0 since in this case vx p x 0 when x 0. Considering D Ä 1 p Ä the case of v > 0, we see that function v.x/ p x vx is convex and differentiable on WD 2.8. FENCHEL CONJUGATES 79 p 1 1=.p 1/ .0; / with .x/ x v. us .x/ 0 if and only if x v , and so v attains 1 v0 D v0 D D its minimum at x v1=.p 1/. erefore, the conjugate function is calculated by D
1 p=.p 1/ n f .v/ 1 Áv ; v R : D p 2
Taking q from q 1 1 p 1, we express the conjugate function as D 80 if v 0; q Ä f .v/ Note that the calculation in Example 2.73(ii) shows that xp vq vx for any x; v 0: Ä p C q e first assertion of the next proposition demonstrates that such a relationship in a more general setting. To formulate the second assertion below, we define the biconjugate of f as the conjugate of f , i.e., f .x/ .f / .x/. WD Proposition 2.74 Let f Rn R be a function with dom f . en we have: W ! ¤ ; (i) v; x f .x/ f .v/ for all x; v Rn. h i Ä C 2 (ii) f .x/ f .x/ for all x Rn. Ä 2 Proof. Observe first that (i) is obvious if f .x/ . If x dom f , we get from (2.45) that D 1 2 f .v/ v; x f .x/, which verifies (i). It implies in turn that h i n n sup ˚ v; x f .v/ ˇ v R « f .x/ for all x; v R ; h i ˇ 2 Ä 2 which thus verifies (ii) and completes the proof. e following important result reveals a close relationship between subgradients and Fenchel conjugates of convex functions. eorem 2.75 For any convex function f Rn R and any x dom f , we have that v W ! N 2 2 @f.x/ if and only if N f .x/ f .v/ v; x : (2.47) N C D h Ni 80 2. SUBDIFFERENTIAL CALCULUS Proof. Taking any v @f.x/ and using definition (2.13) gives us 2 N f .x/ v; x f .x/ v; x for all x Rn : N C h i Ä h Ni 2 is readily implies the inequality n f .x/ f .v/ f .x/ sup ˚ v; x f .x/ ˇ x R « v; x : N C D N C h i ˇ 2 Ä h Ni Since the opposite inequality holds by Proposition 2.74(i), we arrive at (2.47). Conversely, suppose that f .x/ f .v/ v; x . Applying Proposition 2.74(i), we get the N C D h Ni estimate f .v/ v; x f .x/ for every x Rn. is shows that v @f.x/. h i 2 2 N e result obtained allows us to find conditions ensuring that the biconjugate f of a convex function agrees with the function itself. Proposition 2.76 Let x dom f for a convex function f Rn R. Suppose that @f.x/ . en N 2 W ! N ¤ ; we have the equality f .x/ f .x/. N D N Proof. By Proposition 2.74(ii) it suffices to verify the opposite inequality therein. Fix v @f.x/ 2 N and get v; x f .x/ f .v/ by the preceding theorem. is shows that h Ni D N C n f .x/ v; x f .v/ sup ˚ x; v f .v/ ˇ v R « f .x/; N D h Ni Ä h N i ˇ 2 D N which completes the proof of this proposition. Taking into account Proposition 2.47, we arrive at the following corollary. Corollary 2.77 Let f Rn R be a convex function and let x int.dom f /. en we have the W ! N 2 equality f .x/ f .x/ . N D N Finally in this section, we prove a necessary and sufficient condition for the validity of the biconjugacy equality f f known as the Fenchel-Moreau theorem. D eorem 2.78 Let f Rn R be a function with dom f and let A be the set of all affine W ! ¤ ; functions of the form '.x/ a; x b for x Rn, where a Rn and b R. Denote D h i C 2 2 2 A.f / ˚' A ˇ '.x/ f .x/ for all x Rn «: WD 2 ˇ Ä 2 en A.f / whenever epi f is closed and convex. Moreover, the following are equivalent: ¤ ; (i) epi f is closed and convex. (ii) f .x/ sup '.x/ for all x Rn. D ' A.f / 2 (iii) f .x/ f2 .x/ for all x Rn. D 2 2.8. FENCHEL CONJUGATES 81 Proof. Let us first show that A.f / . Fix any x0 dom f and choose 0 < f .x0/. en ¤ ; 2 n .x0; 0/ epi f . By Proposition 2.1, there exist .v; / R R and > 0 such that … N N 2 v; x < v; x0 0 whenever .x; / epi f: (2.48) h N i C N h N i C N 2 Since .x0; f .x0/ ˛/ epi f for all ˛ 0, we get C 2 .f .x/ ˛/ < 0 whenever ˛ 0: N N C N is implies < 0 since if not, we can let ˛ and arrive at a contradiction. For any x N ! 1 2 dom f , it follows from (2.48) as .x; f .x// epi f that 2 v; x f .x/ < v; x0 0 for all x dom f: h N i C N h N i C N 2 is allows us to conclude that v f .x/ > N ; x0 x 0 if x dom f: h i C 2 N N v Define now '.x/ N ; x0 x 0 . en ' A.f /, and so A.f / . WD h i C 2 ¤ ; N N Let us next prove that (i) (ii). By definition we need to show that for any 0 < f .x0/ H) there is ' A.f / such that 0 < '.x0/. Since .x0; 0/ epi f , we apply again Proposition 2.1 2 … to obtain (2.48). In the case where x0 dom f , it was proved above that ' A.f /. Moreover, 2 2 we have '.x0/ 0 < 0 since < 0. Consider now the case where x0 dom f . It follows D N … from (2.48) by taking anyN x dom f and letting that 0. If < 0, we can apply the 2 ! 1 N Ä N same procedure and arrive at the conclusion. Hence we only need to consider the case where 0. In this case N D v; x x0 < 0 whenever x dom f: h N i C 2 Since A.f / , choose '0 A.f / and define ¤ ; 2 'k.x/ '0.x/ k. v; x x0 /; k N : WD C h N i C 2 It is obvious that 'k A.f / and 'k.x0/ '0.x0/ k > 0 for large k. is justifies (ii). 2 D C Let us now verify implication (ii) (iii). Fix any ' A.f /. en ' f , and hence ' f . H) 2 Ä Ä Applying Proposition 2.76 ensures that ' ' f . It follows therefore that D Ä n f .x/ sup ˚'.x/ ˇ ' A.f / f for every x R : D ˇ 2 g Ä 2 e opposite inequality f f holds by Proposition 2.74(ii), and thus f f . Ä D e last implication (iii) (i) is obvious because the set epi g is always closed and convex H) for any function g Rn R. W ! 82 2. SUBDIFFERENTIAL CALCULUS 2.9 DIRECTIONAL DERIVATIVES Our next topic is directional differentiability of convex functions and its relationships with subdif- ferentiation. In contrast to classical analysis, directional derivative constructions in convex analysis are one-sided and related to directions with no classical plus-minus symmetry. Definition 2.79 Let f Rn R be an extended-real-valued function and let x dom f . e W ! N 2 of the function f at the point x in the direction d Rn is the following N 2 limit—if it exists as either a real number or : ˙1 f .x td/ f .x/ f 0.x d/ lim N C N : (2.49) NI WD t 0 t ! C Note that construction (2.49) is sometimes called the right directional derivative f at x in N the direction d. Its left counterpart is defined by f .x td/ f .x/ f 0 .x d/ lim N C N : NI WD t 0 t ! It is easy to see from the definitions that n f 0 .x d/ f 0.x d/ for all d R ; NI D NI 2 and thus properties of the left directional derivative f .x d/ reduce to those of the right one 0 NI (2.49), which we study in what follows. Lemma 2.80 Given a convex function f Rn R with x dom f and given d Rn, define W ! N 2 2 f .x td/ f .x/ '.t/ N C N ; t > 0: WD t en the function ' is nondecreasing on .0; /. 1 Proof. Fix any numbers 0 < t1 < t2 and get the representation t1 t1 x t1d x t2dÁ 1 Áx: N C D t2 N C C t2 N It follows from the convexity of f that t1 t1 f .x t1d/ f .x t2d/ 1 Áf .x/; N C Ä t2 N C C t2 N which implies in turn the inequality f .x t1d/ f .x/ f .x t2d/ f .x/ '.t1/ N C N N C N '.t2/: D t1 Ä t2 D is verifies that ' is nondecreasing on .0; /. 1 2.9. DIRECTIONAL DERIVATIVES 83 e next proposition establishes the directional differentiability of convex functions. Proposition 2.81 For any convex function f Rn R and any x dom f , the directional W ! N 2 derivative f .x d/ .and hence its left counterpart/ exists in every direction d Rn. Furthermore, 0 NI 2 it admits the representation via the function ' is defined in Lemma 2.80: n f 0.x d/ inf '.t/; d R NI D t>0 2 Proof. Lemma 2.80 tells us that the function ' is nondecreasing. us we have f .x td/ f .x/ f 0.x d/ lim N C N lim '.t/ inf '.t/; NI D t 0 t D t 0 D t>0 ! C ! C which verifies the results claimed in the proposition. Corollary 2.82 If f Rn R is a convex function, then f .x d/ is a real number for any x W ! 0 NI N 2 int.dom f / and d Rn. 2 Proof. It follows from eorem 2.29 that f is locally Lipschitz continuous around x, i.e., there N is ` 0 such that ˇf .x td/ f .x/ˇ `t d ˇ N C N ˇ k k ` d for all small t > 0; ˇ t ˇ Ä t D k k which shows that f .x d/ ` d < . j 0 NI j Ä k k 1 To establish relationships between directional derivatives and subgradients of general con- vex functions, we need the following useful observation. Lemma 2.83 For any convex function f Rn R and x dom f , we have W ! N 2 n f 0.x d/ f .x d/ f .x/ whenever d R : NI Ä N C N 2 Proof. Using Lemma 2.80, we have for the function ' therein that '.t/ '.1/ f .x d/ f .x/ for all t .0; 1/; Ä D N C N 2 which justifies the claimed property due to f .x d/ inft>0 '.t/ '.1/. 0 NI D Ä eorem 2.84 Let f Rn R be convex with x dom f . e following are equivalent: W ! N 2 (i) v @f.x/. 2 N n (ii) v; d f 0.x d/ for all d R . h i Ä NI 2 n (iii) f 0 .x d/ v; d f 0.x d/ for all d R . NI Ä h i Ä NI 2 84 2. SUBDIFFERENTIAL CALCULUS Proof. Picking any v @f.x/ and t > 0, we get 2 N v; td f .x td/ f .x/ whenever d Rn; h i Ä N C N 2 which verifies the implication (i) (ii) by taking the limit as t 0 . Assuming now that as- H) ! C sertion (ii) holds, we get by Lemma 2.83 that n v; d f 0.x d/ f .x d/ f .x/ for all d R : h i Ä NI Ä N C N 2 It ensures by definition (2.13) that v @f.x/, and thus assertions (i) and (ii) are equivalent. 2 N It is obvious that (iii) yields (ii). Conversely, if (ii) is satisfied, then for d Rn we have 2 v; d f .x d/, and thus h i Ä 0 NI n f 0 .x d/ f 0.x d/ v; d for any d R : NI D NI Ä h i 2 is justifies the validity of (iii) and completes the proof of the theorem. Let us next list some properties of (2.49) as a function of the direction. Proposition 2.85 For any convex function f Rn R with x dom f , we define the directional W ! N 2 function .d/ f .x d/, which satisfies the following properties: WD 0 NI (i) .0/ 0. D n (ii) .d1 d2/ .d1/ .d2/ for all d1; d2 R . C Ä C 2 (iii) .˛d/ ˛ .d/ whenever d Rn and ˛ > 0. D 2 (iv) If furthermore x int.dom f /, then is finite on Rn. N 2 Proof. It is straightforward to deduce properties (i)–(iii) directly from definition (2.49). For in- stance, (ii) is satisfied due to the relationships f .x t.d1 d2// f .x/ .d1 d2/ lim N C C N C D t 0 t ! C x 2td1 x 2td2 f . N C CN C / f .x/ lim 2 N D t 0 t ! C f .x 2td1/ f .x/ f .x 2td2/ f .x/ lim N C N lim N C N .d1/ .d2/: Ä t 0 2t C t 0 2t D C ! C ! C us it remains to check that .d/ is finite for every d Rn when x int.dom f /. To proceed, 2 N 2 choose ˛ > 0 so small that x ˛d dom f . It follows from Lemma 2.83 that N C 2 .˛d/ f 0.x ˛d/ f .x ˛d/ f .x/ < : D NI Ä N C N 1 Employing (iii) gives us .d/ < . Further, we have from (i) and (ii) that 1 0 .0/ d . d/ .d/ . d/; d Rn; D D C Ä C 2 which implies that .d/ . d/. is yields .d/ > and so verifies (iv). 1 2.9. DIRECTIONAL DERIVATIVES 85 To derive the second major result of this section, yet another lemma is needed. Lemma 2.86 We have the following relationships between a convex function f Rn R and the W ! directional function defined in Proposition 2.85: (i) @f.x/ @ .0/. N D n (ii) .v/ ı˝ .v/ for all v R , where ˝ @ .0/. D 2 WD Proof. It follows from eorem 2.84 that v @f.x/ if and only if 2 N n v; d 0 v; d f 0.x d/ .d/ .d/ .0/; d R : h i D h i Ä NI D D 2 is is equivalent to v @ .0/, and hence (i) holds. 2 To justify (ii), let us first show that .v/ 0 for all v ˝ @ .0/. Indeed, we have D 2 D n .v/ sup ˚ v; d .d/ ˇ d R « v; 0 .0/ 0: D h i ˇ 2 h i D Picking now any v @ .0/ gives us 2 v; d v; d 0 .d/ .0/ .d/; d Rn; h i D h i Ä D 2 which implies therefore that n .v/ sup ˚ v; d .d/ ˇ d R « 0 D h i ˇ 2 Ä and so ensures the validity of .v/ 0 for any v @ .0/. D 2 n It remains to verify that .v/ if v @ .0/. For such an element v, find d0 R D 1 … 2 with v; d0 > .d0/. Since is positively homogeneous by Proposition 2.85, it follows that h i n .v/ sup ˚ v; d .d/ ˇ d R « sup. v; td0 .td0// D h i ˇ 2 t>0 h i sup t. v; d0 .d0// ; D t>0 h i D 1 which completes the proof of the lemma. Now we are ready establish a major relationship between the directional derivative and the subdifferential of an arbitrary convex function. eorem 2.87 Given a convex function f Rn R and a point x int.dom f /, we have W ! N 2 n f 0.x d/ max ˚ v; d ˇ v @f.x/« for any d R : NI D h i ˇ 2 N 2 86 2. SUBDIFFERENTIAL CALCULUS Proof. It follows from Proposition 2.76 that n f 0.x d/ .d/ .d/; d R ; NI D D 2 by the properties of the function .d/ f .x d/ from Proposition 2.85. Note that is a finite D 0 NI convex function in this setting. Employing now Lemma 2.86 tells us that .v/ ı˝ .v/, where D ˝ @ .0/ @f.x/. Hence we have by (2.46) that D D N .d/ ı .d/ sup ˚ v; d ˇ v ˝«: D ˝ D h i ˇ 2 Since the set ˝ @f.x/ is compact by Proposition 2.47, we complete the proof. D N 2.10 SUBGRADIENTS OF SUPREMUM FUNCTIONS Let T be a nonempty subset of Rp and let g T Rn R. For convenience, we also use the W n! notation gt .x/ g.t; x/. e supremum function f R R for gt over T is WD W ! n f .x/ sup g.t; x/ sup gt .x/; x R : (2.50) WD t T D t T 2 2 2 If the supremum in (2.50) is attained (this happens, in particular, when the index set T is compact and g. ; x/ is continuous), then (2.50) reduces to the maximum function, which can be written in form (2.35) when T is a finite set. e main goal of this section is to calculate the subdifferential (2.13) of (2.50) when the functions gt are convex. Note that in this case the supremum function (2.50) is also convex by n Proposition 1.43. In what follows we assume without mentioning it that the functions gt R R W ! are convex for all t T . 2 For any point x Rn, define the active index set N 2 S.x/ ˚t T ˇ gt .x/ f .x/«; (2.51) N WD 2 ˇ N D N which may be empty if the supremum in (2.50) is not attained for x x. We first present a simple DN lower subdifferential estimate for (2.66). Proposition 2.88 Let dom f for the supremum function (2.50) over an arbitrary index set T . ¤ ; For any x dom f , we have the inclusion N 2 clco [ @gt .x/ @f.x/: (2.52) N N t S.x/ 2 N Proof. Inclusion (2.52) obviously holds if S.x/ . Supposing now that S.x/ , fix t S.x/ N D; N ¤ ; 2 N and v @gt .x/. en we get gt .x/ f .x/ and therefore 2 N N D N v; x x gt .x/ gt .x/ gt .x/ f .x/ f .x/ f .x/; h Ni Ä N D N Ä N which shows that v @f.x/. Since the subgradient set @f.x/ is a closed and convex, we conclude 2 N N that inclusion (2.52) holds in this case. 2.10. SUBGRADIENTS OF SUPREMUM FUNCTIONS 87 Further properties of the supremum/maximum function (2.50) involve the following useful extensions of continuity. Its lower version will be studied and applied in Chapter 4. Definition 2.89 Let ˝ be a nonempty subset of Rn. A function f ˝ R is - W ! x ˝ if for every sequence xk in ˝ converging to x it holds N 2 f g N lim sup f .xk/ f .x/: k Ä N !1 We say that f is ˝ if it is upper semicontinuous at every point of this set. Proposition 2.90 Let T be a nonempty, compact subset of Rp and let g. x/ be upper semicontinuous I on T for every x Rn. en f in (2.50) is a finite-valued convex function. 2 Proof. We need to check that f .x/ < for every x Rn. Suppose by contradiction that f .x/ n 1 2 N D for some x R . en there exists a sequence tk T such that g.tk; x/ . Since T 1 N 2 f g N ! 1 is compact, assume without loss of generality that tk t T . By the upper semicontinuity of ! N 2 g. ; x/ on T we have N lim sup g.tk; x/ g.t; x/ < ; 1 D k N Ä N N 1 !1 which is a contradiction that completes the proof of the proposition. Proposition 2.91 Let the index set T Rp be compact and let g. x/ be upper semicontinuous ; ¤ I for every x Rn. en the active index set S.x/ T is nonempty and compact for every x Rn. 2 N N 2 Furthermore, we have the compactness in Rn of the convex hull C co [ @gt .x/: WD N t S.x/ 2 N Proof. Fix x Rn and get from Proposition 2.90 that f .x/ < for the supremum function N 2 N 1 (2.50). Consider an arbitrary sequence tk T with g.tk; x/ f .x/ as k . By the com- f g N ! N ! 1 pactness of T we find t T such that tk t along some subsequence. It follows from the as- N 2 ! N sumed upper semicontinuity of g that f .x/ lim sup g.tk; x/ g.t; x/ f .x/; N D k N Ä N N Ä N !1 which ensures that t S.x/ by the construction of S in (2.51), and thus S.x/ . Since g.t; x/ N 2 N N ¤ ; N Ä f .x/, for any t T we get the representation N 2 S.x/ ˚t T ˇ g.t; x/ f .x/« N D 2 ˇ N N 88 2. SUBDIFFERENTIAL CALCULUS implying the closedness of S.x/ (and hence its compactness since S.x/ T ), which is an imme- N N diate consequence of the upper semicontinuity of g. x/. I N It remains to show that the subgradient set Q [ @gt .x/ WD N t S.x/ 2 N is compact in Rn; this immediately implies the claimed compactness of C co Q, since the D convex hull of a compact set is compact (show it, or see Corollary 3.7 in the next chapter). To proceed with verifying the compactness of Q, we recall first that Q @f.x/ by Proposition 2.88. N is implies that the set Q is bounded in Rn, since the subgradient set of the finite-valued convex function is compact by Proposition 2.47. To check the closedness of Q, take a sequence vk Q f g converging to some v and for each k N and find tk S.x/ such that vk @gt .x/. Since S.x/ N 2 2 N 2 k N N is compact, we may assume that tk t S.x/. en g.tk; x/ g.t; x/ f .x/ and ! N 2 N N D N N D N n vk; x x g.tk; x/ g.tk; x/ g.tk; x/ g.t; x/ for all x R h Ni Ä N D N N 2 by definition (2.13). is gives us the relationships v; x x lim sup vk; x x lim sup g.tk; x/ g.t; x/ g.t; x/ g.t; x/ gt .x/ gt .x/; h N Ni D k h Ni Ä k N N Ä N N N D N N N !1 !1 which verify that v @gt .x/ Q and thus complete the proof. N 2 N N e next result is of its own interest while being crucial for the main subdifferential result of this section. eorem 2.92 In the setting of Proposition 2.91 we have n n f 0.x v/ C .v/ for all x R ; v R NI Ä N 2 2 via the support function of the set C defined therein. Proof. It follows from Proposition 2.81 and Proposition 2.82 that < f .x v/ 1 0 NI D inf>0 './ < , where the function ' is defined in Lemma 2.80 and is proved to be nonde- 1 creasing on .0; /. us there is a strictly decreasing sequence k 0 as k with 1 # ! 1 f .x kv/ f .x/ f 0.x v/ lim N C N ; k N : NI D k k 2 !1 For every k, we select tk T such that f .x kv/ g.tk; x kv/. e compactness of T 2 N C D N C allows us to find t T such that tk t T along some subsequence. Let us show that t S.x/. N 2 ! N 2 N 2 N Since g.tk; / is convex and k < 1 for large k, we have g.tk; x kv/ g tk; k.x v/ .1 k/x kg.tk; x v/ .1 k/g.tk; x/; (2.53) N C D N C C N Ä N C C N 2.10. SUBGRADIENTS OF SUPREMUM FUNCTIONS 89 which implies by the continuity of f (any finite-valued convex function is continuous) and the upper semicontinuity of g. x/ that I f .x/ lim f .x kv/ lim g.tk; x kv/ g.t; x/: N D k N C D k N C Ä N N !1 !1 is yields f .x/ g.t; x/, and so t S.x/. Moreover, for any > 0 and large k, we get N D N N N 2 N g.tk; x v/ with g.t; x v/ N C Ä C WD N N C by the upper semicontinuity of g. It follows from (2.53) that g.tk; x kv/ k. / g.tk; x/ N C C f .x/; N 1 k ! N which tells us that limk g.tk; x/ f .x/ due to !1 N D N f .x/ lim inf g.tk; x/ lim sup g.tk; x/ g.t; x/ f .x/: N Ä k N Ä k N Ä N N D N !1 !1 Fix further any > 0 and, taking into account that k < for all k sufficiently large, deduce from Lemma 2.80 that f .x kv/ f .x/ g.tk; x kv/ g.t; x/ N C N N C N N k D k g.tk; x kv/ g.tk; x/ N C N Ä k g.tk; x v/ g.tk; x/ N C N Ä is ensures in turn the estimates f .x kv/ f .x/ g.tk; x v/ g.tk; x/ lim sup N C N lim sup N C N k k Ä k !1 g.!1t; x v/ g.t; x/ N N C N N Ä gt .x v/ gt .x/ N N C N N ; D which implies the relationships gt .x v/ gt .x/ f 0.x v/ inf N N C N N gt0 .x v/: NI Ä >0 D N NI Using finally eorem 2.87 and the inclusion @gt .x/ C from Proposition 2.88 give us N N f 0.x v/ gt0 .x v/ sup w; v sup w; v C .v/; NI Ä N NI D h i Ä h i D w @gt .x/ w C 2 N N 2 and justify therefore the statement of the theorem. 90 2. SUBDIFFERENTIAL CALCULUS Now we are ready to establish the main result of this section. eorem 2.93 Let f be defined in (2.50), where the index set T Rp is compact, and ; ¤ where the function g.t; / is convex on Rn for every t T . Suppose in addition that the function 2 g. ; x/ is upper semicontinuous on T for every x Rn. en we have 2 @f.x/ co [ @gt .x/: N D N t S.x/ 2 N Proof. Taking any w @f.x/, we deduce from eorem 2.87 and eorem 2.92 that 2 N n w; v f 0.x; v/ C .v/ for all v R ; h i Ä N Ä 2 where C is defined in Proposition 2.91. us w @C .0/ C by formula (2.44) for the set 2 D ˝ C , which is a nonempty, closed, convex subset of Rn. e opposite inclusion follows from D Proposition 2.88. 2.11 EXERCISES FOR CHAPTER 2 Exercise 2.1 (i) Let ˝ Rn be a nonempty, closed, convex cone and let x ˝. Show that N … there exists a nonzero element v Rn such that v; x 0 for all x ˝ and v; x > 0. 2 h i Ä 2 h Ni (ii) Let ˝ be a subspace of Rn and let x ˝. Show that there exists a nonzero element v Rn N … 2 such that v; x 0 for all x ˝ and v; x > 0. h i D 2 h Ni n Exercise 2.2 Let ˝1 and ˝2 be two nonempty, convex subsets of R satisfying the condition n ri ˝1 ri ˝2 . Show that there is a nonzero element v R which separates the sets ˝1 and \ D; 2 ˝2 in the sense of (2.2). Exercise 2.3 Two nonempty, convex subsets of Rn are called properly separated if there exists a nonzero element v Rn such that 2 sup ˚ v; x ˇ x ˝1« inf ˚ v; y ˇ y ˝2«; h i ˇ 2 Ä h i ˇ 2 inf ˚ v; x ˇ x ˝1« < sup ˚ v; y ˇ y ˝2«: h i ˇ 2 h i ˇ 2 Prove that ˝1 and ˝2 are properly separated if and only if ri ˝1 ri ˝2 . \ D; Exercise 2.4 Let f Rn R be a convex function attaining its minimum at some point x W ! n N 2 dom f . Show that the sets ˝1; ˝2 with ˝1 epi f and ˝2 R f .x/ form an extremal f g WD WD f N g system. 2.11. EXERCISES FOR CHAPTER 2 91 n Exercise 2.5 Let ˝i for i 1; 2 be nonempty, convex subsets of R with at least one point in D common and satisfy int ˝1 and int ˝1 ˝2 : ¤ ; \ D; n Show that there are elements v1; v2 in R , not equal to zero simultaneously, and real numbers ˛1; ˛2 such that vi ; x ˛i for all x ˝i ; i 1; 2; h i Ä 2 D v1 v2 0; and ˛1 ˛2 0: C D C D Exercise 2.6 Let ˝ x Rn ˇ x 0 . Show that WD f 2 ˇ Ä g N.x ˝/ ˚v Rn ˇ v 0; v; x 0« for any x ˝: NI D 2 ˇ h Ni D N 2 Exercise 2.7 Let f Rn R be a convex function differentiable at x. Show that W ! N N .x; f .x// epi f ˚ f .x/; 1 ˇ 0«: N N I D r N ˇ Exercise 2.8 Use induction to prove Corollary 2.19. Exercise 2.9 Calculate the subdifferentials and the singular subdifferentials of the following convex functions at every point of their domains: (i) f .x/ x 1 x 2 ; x R. D j j C j j 2 (ii) f .x/ e x ; x R. D j j 2 (x2 1 if x 1; (iii) f .x/ j j Ä D otherwise on R : 1 ( p1 x2 if x 1; (iv) f .x/ j j Ä D otherwise on R : 1 2 (v) f .x1; x2/ x1 x2 ; .x1; x2/ R . D j j C j j 2 2 (vi) f .x1; x2/ max ˚ x1 ; x2 «; .x1; x2/ R . D j j j j 2 (vii) f .x/ max ˚ a; x b; 0«; x Rn, where a Rn and b R. D h i 2 2 2 Exercise 2.10 Let f R R be a nondecreasing convex function and let x R. Show that W ! N 2 v 0 for v @f.x/. 2 N Exercise 2.11 Use induction to prove Corollary 2.46. Exercise 2.12 Let f Rn R be a convex function. Give an alternative proof for the part of W ! Proposition 2.47 stating that @f.x/ for any element x int.dom f /. Does the conclusion N ¤ ; N 2 still holds if x ri.dom f /? N 2 92 2. SUBDIFFERENTIAL CALCULUS n Exercise 2.13 Let ˝1 and ˝2 be nonempty, convex subsets of R . Suppose that ri ˝1 ri ˝2 \ ¤ . Prove that ; N.x ˝1 ˝2/ N.x ˝1/ N.x ˝2/ for all x ˝1 ˝2: NI \ D NI C NI N 2 \ n Exercise 2.14 Let f1; f2 R R be convex functions satisfying the condition W ! ri .dom f1/ ri .dom f2/ : \ ¤ ; Prove that for all x dom f1 dom f2 we have N 2 \ @.f1 f2/.x/ @f1.x/ @f2.x/; @1.f1 f2/.x/ @1f1.x/ @1f2.x/: C N D N C N C N D N C N Exercise 2.15 In the setting of eorem 2.51 show that N..x; y; z/ ˝1/ Œ N..x; y; z/ ˝2/ N N N I \ N N N I under the qualification condition (2.33). Exercise 2.16 Calculate the singular subdifferential of the composition f B in the setting of ı eorem 2.51. Exercise 2.17 Consider the composition g h in the setting of Proposition 1.54 and take any ı x dom .g h/. Prove that N 2 ı p @.g h/.x/ n X i vi ˇ .1; : : : ; p/ @g.y/; vi @fi .x/; i 1; : : : ; po ı N D ˇ 2 N 2 N D i 1 ˇ D under the validity of the qualification condition p h0 X i @fi .x/; .1; : : : ; p/ @1g.y/i i 0 for i 1; : : : ; p: 2 N 2 N H) D D i 1 D Exercise 2.18 Calculate the singular subdifferential of the optimal value functions in the setting of eorem 2.61. Exercise 2.19 Calculate the support functions for the following sets: n n (i) ˝1 ˚x .x1; : : : ; xn/ R ˇ x 1 Pi 1 xi 1«: WD D 2 nˇ k k WD D nj j Ä (ii) ˝2 ˚x .x1; : : : ; xn/ R ˇ x maxi 1 xi 1«: WD D 2 ˇ k k1 WD D j j Ä 2.11. EXERCISES FOR CHAPTER 2 93 Exercise 2.20 Find the Fenchel conjugate for each of the following functions on R: (i) f .x/ ex. D ln.x/ x > 0; (ii) f .x/ D x 0: 1 Ä (iii) f .x/ ax2 bx c, where a > 0. D C C (iv) f .x/ ıIB .x/, where IB Œ 1; 1. D D (v) f .x/ x . D j j n Exercise 2.21 Let ˝ be a nonempty subset of R . Find the biconjugate function ı˝. Exercise 2.22 Prove the following statements for the listed functions on Rn: v (i) .f / .v/ f Á, where > 0. D (ii) .f /.v/ f .v/ , where R. C D 2 n (iii) f .v/ f .v/ u; v , where u R and fu.x/ f .x u/. u D C h i 2 WD Exercise 2.23 Let f Rn R be a convex and positively homogeneous function. Show that W !n f .v/ ı˝ .v/ for all v R , where ˝ @f.0/. D 2 WD Exercise 2.24 Let f Rn R be a function with dom f . Show that f .x/ f .x/ for W ! D; D all x Rn. 2 Exercise 2.25 Let f R R be convex. Prove that @f.x/ Œf 0 .x/; f 0 .x/, where f 0 .x/ W ! D C and f 0 .x/ stand for the left and right derivative of f at x, respectively. C Exercise 2.26 Define the function f R R by W ! ( p1 x2 if x 1; f .x/ j j Ä WD otherwise: 1 Show that f is a convex function and calculate its directional derivatives f . 1 d/ and f .1 d/ 0 I 0 I in the direction d 1. D Exercise 2.27 Let f Rn R be convex and let x dom f . For an element d Rn, consider W ! N 2 2 the function ' defined in Lemma 2.80 and show that ' is nondecreasing on . ; 0/. 1 Exercise 2.28 Let f Rn R be convex and let x dom f . Prove the following: W ! N 2 n (i) f 0 .x d/ f 0.x d/ for all d R . NI Ä NI 2 n (ii) f .x/ f .x d/ f 0 .x d/ f 0.x d/ f .x d/ f .x/, d R . N N Ä NI Ä NI Ä N C N 2 Exercise 2.29 Prove the assertions of Proposition 2.67. 94 2. SUBDIFFERENTIAL CALCULUS Exercise 2.30 Let ˝ Rn be a nonempty, compact set. Using eorem 2.93, calculate the subdifferential of the support function ˝ .0/. Exercise 2.31 Let ˝ be a nonempty, compact subset of Rn. Define the function n ˝ .x/ sup ˚ x ! ˇ ! ˝ ; x R : WD k k ˇ 2 g 2 n Prove that ˝ is a convex function and find its subdifferential @˝ .x/ at any x R . 2 Exercise 2.32 Using the representation x sup ˚ v; x ˇ v 1« k k D h i ˇ k k Ä and eorem 2.93, calculate the subdifferential @p.x/ of the norm function p.x/ x at any N WD k k point x Rn. N 2 Exercise 2.33 Let d.x ˝/ be the distance function (1.5) associated with some nonempty, I closed, convex subset ˝ of Rn. (i) Prove the representation d.x ˝/ sup ˚ x; v ˝ .v/ ˇ v 1« I D h i ˇ k k Ä via the support function of ˝. (ii) Calculate the subdifferential @d.x ˝/ at any point x Rn. NI N 2 95 C H A P T E R 3 Remarkable Consequences of Convexity is chapter is devoted to remarkable topics of convex analysis that, together with subdifferential calculus and related results of Chapter 2, constitute the major achievements of this discipline most important for many applications—first of all to convex optimization. 3.1 CHARACTERIZATIONS OF DIFFERENTIABILITY In this section we define another classical notion of differentiability, the Gâteaux derivative, and show that for convex functions it can be characterized, together with the Fréchet one, via a single- element subdifferential of convex analysis. Definition 3.1 Let f Rn R be finite around x. We say that f is G W ! N at x if there is v Rn such that for any d Rn we have N 2 2 f .x td/ f .x/ t v; d lim N C N h i 0: t 0 t D ! en we call v the G of f at x and denote by f .x/. N G0 N It follows from the definition and the equality f .x d/ f .x d/ that f .x/ v if 0 NI D 0 NI G0 N D and only if the directional derivative f .x d/ exists and is linear with respect to d with f .x d/ 0 NI 0 NI D v; d for all d Rn; see Exercise 3.1 and its solution. Moreover, if f is Gâteaux differentiable h i 2 at x, then N f .x td/ f .x/ n fG0 .x/; d lim N C N for all d R : h N i D t 0 t 2 ! Proposition 3.2 Let f Rn R be a function (not necessarily convex) which is locally Lipschitz W ! continuous around x dom f . en the following assertions are equivalent: N 2 (i) f is Fréchet differentiable at x. N (ii) f is Gâteaux differentiable at x. N 96 3. REMARKABLE CONSEQUENCES OF CONVEXITY Proof. Let f be Fréchet differentiable at x and let v f .x/. For a fixed 0 d Rn, taking N WD r N ¤ 2 into account that x td x t d 0 as t 0, we have k N C Nk D j j k k ! ! f .x td/ f .x/ t v; d f .x td/ f .x/ v; td lim N C N h i d lim N C N h i 0; t 0 t D k k t 0 t d D ! ! k k which implies the Gateaux differentiablity of f at x with f .x/ f .x/. Note that the local O N G0 N D r N Lipschitz continuity of f around x is not required in this implication. N Assuming now that f is Gateaux differentiable at x with v f .x/, let us show that O N WD G0 N f .x h/ f .x/ v; h lim N C N h i 0: h 0 h D ! k k Suppose by contradiction that it does not hold. en we can find 0 > 0 and a sequence of hk 0 ! with hk 0 as k N such that ¤ 2 ˇf .x hk/ f .x/ v; hk ˇ N C N h ij 0: (3.1) hk k k hk Denote tk hk and dk . en tk 0, and we can assume without loss of generality WD k k WD hk ! k k that dk d with d 1. is gives us the estimate ! k k D ˇf .x tkdk/ f .x/ v; tkdk 0 dk ˇ N C N h ij: k k Ä tk Let ` 0 be a Lipschitz constant of f around x. e triangle inequality ensures that N ˇf .x tkdk/ f .x/ v; tkdk 0 dk ˇ N C N h ij k k Ä tk ˇf .x tkdk/ f .x tkd/ v; tkd v; tkdk ˇf .x tkd/ f .x/ v; tkd ˇ N C N C j C jh i h ij ˇ N C N h ij Ä tk C tk ˇf .x tkd/ f .x/ v; tkd ` dk d v dk d ˇ N C N h ij 0 as k : Ä k k C k k k k C tk ! ! 1 is contradicts (3.1) since 0 dk 0, so f is Fréchet differentiable at x. k k ! N Now we are ready to derive a nice subdifferential characterization of differentiability. eorem 3.3 Let f Rn R be a convex function with x int(domf /. e following asser- W ! N 2 tions are equivalent: (i) f is Fréchet differentiable at x. N (ii) f is Gâteaux differentiable at x. N (iii) @f.x/ is a singleton. N 3.2. CARATHÉODORY THEOREM AND FARKAS LEMMA 97 Proof. It follows from eorem 2.29 that f is locally Lipschitz continuous around x, and N hence (i) and (ii) are equivalent by Proposition 3.2. Suppose now that (ii) is satisfied and let v f .x/. en f .x d/ f .x d/ v; d for all d Rn. Observe that @f.x/ since WD G0 N 0 NI D 0 NI D h i 2 N ¤ ; x int(domf /. Fix any w @f.x/. It follows from eorem 2.84 that N 2 2 N n f 0 .x d/ w; d f 0.x d/ whenever d R ; NI Ä h i Ä NI 2 and thus w; d v; d , so w v; d 0 for all d. Plugging there d w v gives us w v h i D h i h i D D D and thus justifies that @f.x/ is a singleton. N Suppose finally that (iii) is satisfied with @f.x/ v and then show that (ii) holds. By N D f g eorem 2.87 we have the representation f 0.x d/ sup w; d v; d ; NI D w @f.x/h i D h i 2 N for all d Rn. is implies that f is Gateaux and hence Fréchet differentiable at x. 2 O N 3.2 CARATHÉODORY THEOREM AND FARKAS LEMMA We begin with preliminary results on conic and convex hulls of sets that lead us to one of the most remarkable results of convex geometry established by Constantin Carathéodory in 1911. Recall that a set ˝ is called a cone if x ˝ for all 0 and x ˝. It follows from the definition that 2 2 n 0 ˝ if ˝ is a nonempty cone. Given a nonempty set ˝ R , we say that K˝ is the convex 2 cone generated by ˝, or the convex conic hull of ˝, if it is the intersection of all the convex cones containing ˝. Figure 3.1: Convex cone generated by a set. 98 3. REMARKABLE CONSEQUENCES OF CONVEXITY Proposition 3.4 Let ˝ be a nonempty subset of Rn. en the convex cone generated by ˝ admits the following representation: m K˝ n X i ai ˇ i 0; ai ˝; m N o: (3.2) D ˇ 2 2 i 1 ˇ D If, in particular, the set ˝ is convex, then K˝ R ˝. D C Proof. Denote by C the set on the right-hand side of (3.2) and observe that it is a convex cone which contains ˝. It remains to show that C K for any convex cone K containing ˝. To m see it, fix such a cone and form the combination x Pi 1 i ai with any i 0, ai ˝, and D D 2 m N. If i 0 for all i, then x 0 K. Otherwise, suppose that i > 0 for some i and denote 2 m D D 2 Pi 1 i > 0. is tells us that WD D m i x X ai Á K; D 2 i 1 D which yields C K, so K˝ C . e last assertion of the proposition is obvious. D n Proposition 3.5 For any nonempty set ˝ R and any x K˝ 0 , we have 2 n f g m x X i ai with i > 0; ai ˝ as i 1; : : : ; m and m n: D 2 D Ä i 1 D m Proof. Let x K˝ 0 . It follows from Proposition 3.4 that x Pi 1 i ai with some i > 0 2 n f g D D and ai ˝ as for i 1; : : : ; m and m N. If the elements a1; : : : ; am are linearly dependent, 2 D 2 then there are numbers i R, not all zeros, such that 2 m X i ai 0: D i 1 D We can assume that I i 1; : : : ; m i > 0 and get for any > 0 that WD f D j g ¤ ; m m m m x X i ai X i ai X i ai Á X i i Áai : D D D i 1 i 1 i 1 i 1 D D D D n i ˇ o i0 Let min ˇ i I , where i0 I , and denote ˇi i i for i 1; : : : ; m. WD i ˇ 2 D i0 2 WD D en we have the relationships m x X ˇi ai with ˇi 0 and ˇi 0 for any i 1; : : : ; m: D 0 D D i 1 D 3.2. CARATHÉODORY THEOREM AND FARKAS LEMMA 99 Continuing this procedure, after a finite number of steps we represent x as a positive linear com- bination of linearly independent elements aj j J , where J 1; : : : ; m . us the index f j 2 g f g set J cannot contain more than n elements. e following remarkable result is known as the Carathéodory theorem. It relates convex hulls of sets with the dimension of the space; see an illustration in Figure 3.2. Figure 3.2: Carathéodory theorem. eorem 3.6 Let ˝ be a nonempty subset of Rn. en every x co ˝ can be represented as a 2 convex combination of no more than n 1 elements of the set ˝. C n 1 Proof. Denoting B 1 ˝ R C , it is easy to observe that co B 1 co ˝ and that WD f g D f g co B KB . Take now any x co ˝. en .1; x/ co B and by Proposition 3.5 find i 0 and 2 2 .1; ai / B for i 0; : : : ; m with m n such that 2 D Ä m .1; x/ X i .1; ai /: D i 0 D m m is implies that x Pi 0 i ai with Pi 0 i 1, i 0, and m n. D D D D Ä e next useful assertion is a direct consequence of eorem 3.6. Corollary 3.7 Suppose that a subset ˝ of Rn is compact. en the set co ˝ is also compact. Now we present two propositions, which lead us to another remarkable result known as the Farkas lemma; see eorem 3.10. 100 3. REMARKABLE CONSEQUENCES OF CONVEXITY n Proposition 3.8 Let a1; : : : ; am be linearly dependent elements in R , where ai 0 for all i ¤ D 1; : : : ; m. Define the sets ˝ ˚a1; : : : ; am« and ˝i ˝ ˚ai «; i 1; : : : ; m: WD WD n D en the convex cones generated by these sets are related as m K˝ [ K˝ : (3.3) D i i 1 D m Proof. Obviously, i 1K˝i K˝ . To prove the converse inclusion, pick x K˝ and get [ D 2 m x X i ai with i 0; i 1; : : : ; m: D D i 1 D m If x 0, then x i 1K˝i obviously. Otherwise, the linear dependence of ai allows us to find D 2 [ D i R not all zeros such that 2 m X i ai 0: D i 1 D Following the proof of Proposition 3.5, we represent x as m x X i ai with some i0 1; : : : ; m : D 2 f g i 1;i i0 D ¤ m is shows that x i 1K˝i and thus verifies that equality (3.3) holds. 2 [ D n Proposition 3.9 Let a1; : : : ; am be elements of R . en the convex cone K˝ generated by the set n ˝ a1; : : : ; am is closed in R . WD f g Proof. Assume first that the elements a1; : : : ; am are linearly independent and take any sequence xk K˝ converging to x. By the construction of K˝ , find numbers ˛ki 0 for each i f g D 1; : : : ; m and k N such that 2 m xk X ˛ki ai : D i 1 D m Letting ˛k .˛k1; : : : ; ˛km/ R and arguing by contradiction, it is easy to check that the WD 2 sequence ˛k is bounded. is allows us to suppose without loss of generality that ˛k f g ! .˛1; : : : ; ˛m/ as k with ˛i 0 for all i 1; : : : ; m. us we get by passing to the limit m! 1 D that xk x Pi 1 ˛i ai K˝ , which verifies the closedness of K˝ . ! D D 2 3.2. CARATHÉODORY THEOREM AND FARKAS LEMMA 101 Considering next arbitrary elements a1; : : : ; am, assume without loss of generality that ai ¤ 0 for all i 1; : : : ; m. By the above, it remains to examine the case where a1; : : : ; am are linearly D dependent. en by Proposition 3.8 we have the cone representation (3.3), where each set ˝i contains m 1 elements. If at least one of the sets ˝i consists of linearly dependent elements, we can represent the corresponding cone K˝ in a similar way via the sets of m 2 elements. is i gives us after a finite number of steps that p K˝ [ KC ; D j j 1 D where each set Cj ˝ contains only linear independent elements, and so the cones KC are j closed. us the cone K˝ under consideration is closed as well. Now we are ready to derive the fundamental result of convex analysis and optimization established by Gyula Farkas in 1894. n n eorem 3.10 Let ˝ a1; : : : ; am with a1; : : : ; am R and let b R . e following as- WD f g 2 2 sertions are equivalent: (i) b K˝ . 2 (ii) For any x Rn, we have the implication 2 ai ; x 0; i 1; : : : ; m b; x 0: h i Ä D H) h i Ä Proof. To verify .i/ .ii/, take b K˝ and find i 0 for i 1; : : : ; m such that H) 2 D m b X i ai : D i 1 D n en, given x R , the inequalities ai ; x 0 for all i 1; : : : ; m imply that 2 h i Ä D m b; x X i ai ; x 0; h i D h i Ä i 1 D which is (ii). To prove the converse implication, suppose that b K˝ and show that (ii) does … not hold in this case. Indeed, since the set K˝ is closed and convex, the strict separation result of Proposition 2.1 yields the existence of x Rn satisfying N 2 sup ˚ u; x ˇ u K˝ < b; x : h Ni ˇ 2 g h Ni Since K˝ is a cone, we have 0 0; x < b; x and tu K˝ for all t > 0, u K˝ . us D h Ni h Ni 2 2 t sup ˚ u; x ˇ u K˝ « < b; x whenever t > 0: h Ni ˇ 2 h Ni 102 3. REMARKABLE CONSEQUENCES OF CONVEXITY Dividing both sides of this inequality by t and letting t give us ! 1 sup ˚ u; x ˇ u K˝ « 0: h Ni ˇ 2 Ä It follows that ai ; x 0 for all i 1; : : : ; m while b; x > 0. is is a contradiction, which h Ni Ä D h Ni therefore completes the proof of the theorem. 3.3 RADON THEOREM AND HELLY THEOREM In this section we present a celebrated theorem of convex geometry discovered by Edward Helly in 1913. Let us start with the following simple lemma. n Lemma 3.11 Consider arbitrary elements w1; : : : ; wm in R with m n 2. en these elements C are affinely dependent. n Proof. Form the elements w2 w1; : : : ; wm w1 in R and observe that these elements are lin- early dependent since m 1 > n. us the elements w1; : : : ; wm are affinely dependent by Propo- sition 1.63. We continue with a theorem by Johann Radon established in 1923 and used by himself to give a simple proof of the Helly theorem. n eorem 3.12 Consider the elements w1; : : : ; wm of R with m n 2 and let I C WD 1; : : : ; m . en there exist two nonempty, disjoint subsets I1 and I2 of I with I I1 I2 f g D [ such that for ˝1 wi i I1 , ˝2 wi i I2 one has WD f j 2 g WD f j 2 g co ˝1 co ˝2 : \ ¤ ; Proof. Since m n 2, it follows from Lemma 3.11 that the elements w1; : : : ; wm are affinely C dependent. us there are real numbers 1; : : : ; m, not all zeros, such that m m X i wi 0; X i 0: D D i 1 i 1 D D Consider the index sets I1 i 1; : : : ; m i 0 and I2 i 1; : : : ; m i < 0 . Since m WD f D j g WD f D j g P i 0, both sets I1 and I2 are nonempty and P i P i . Denoting i 1 D i I1 D i I2 WD D 2 2 Pi I1 i , we have the equalities 2 i i X i wi X i wi and X wi X wi : D D i I1 i I2 i I1 i I2 2 2 2 2 3.4. TANGENTS TO CONVEX SETS 103 Define ˝1 wi i I1 , ˝2 wi i I2 and observe that co ˝1 co ˝2 since WD f j 2 g WD f j 2 g \ ¤ ; i i X wi X wi co ˝1 co ˝2: D 2 \ i I1 i I2 2 2 is completes the proof of the Radon theorem. Now we are ready to formulate and prove the fundamental Helly theorem. n eorem 3.13 Let O ˚˝1; : : : ; ˝m« be a collection of convex sets in R with m n 1. WD C Suppose that the intersection of any subcollection of n 1 sets from O is nonempty. en we m C have Ti 1 ˝i . D ¤ ; Proof. Let us prove the theorem by using induction with respect to the number of elements in O. e conclusion of the theorem is obvious if m n 1. Suppose that the conclusion holds for n D C any collection of m convex sets of R with m n 1. Let ˚˝1; : : : ; ˝m; ˝m 1« be a collection C C of convex sets in Rn such that the intersection of any subcollection of n 1 sets is nonempty. For C i 1; : : : ; m 1, define D C m 1 C i \ ˝j WD j 1;j i D ¤ and observe that i ˝j whenever j i and i; j 1; : : : ; m 1. It follows from the in- ¤ D C duction hypothesis that i , and so we can pick wi i for every i 1; : : : ; m 1. Ap- ¤ ; 2 D C plying the above Radon theorem for the elements w1; : : : wm 1, we find two nonempty, dis- C joint subsets index sets I1 and I2 of I 1; : : : ; m 1 with I I1 I2 such that for W1 WD f C g D [ WD wi i I1 and W2 wi i I2 one has co W1 co W2 . is allows us to pick an ele- f j 2 g WD f j 2 g \m 1 ¤ ; ment w co W1 co W2. We finally show that w i C1 ˝i and thus arrive at the conclusion. 2 \ 2 \ D Since i j for i I1 and j I2, it follows that i ˝j whenever i I1 and j I2. Fix ¤ 2 2 2 2 any i 1; : : : ; m 1 and consider the case where i I1. en wj j ˝i for every j I2. D C 2 2 2 Hence w co W2 co wj j I2 ˝i due the convexity of ˝i . It shows that w ˝i for 2 D f j 2 g 2 every i I1. Similarly, we can verify that w ˝i for every i I2 and thus complete the proof 2 2 2 of the theorem. 3.4 TANGENTS TO CONVEX SETS In Chapter 2 we studied the normal cone to convex sets and learned that the very definition and major properties of it were related to convex separation. In this section we define the tangent cone to convex sets independently while observing that it can be constructed via duality from the normal cone to the set under consideration. We start with cones. Given a nonempty, convex cone C Rn, the polar of C is n C ı ˚v R ˇ v; c 0 for all c C «: WD 2 ˇ h i Ä 2 104 3. REMARKABLE CONSEQUENCES OF CONVEXITY Proposition 3.14 For any nonempty, convex cone C Rn, we have .C ı/ı cl C: D Proof. e inclusion C .C / follows directly from the definition, and hence cl C .C / ı ı ı ı since .C / is a closed set. To verify the converse inclusion, take any c .C / and suppose by ı ı 2 ı ı contradiction that c cl C . Since C is a cone, by convex separation from Proposition 2.1, find … v 0 such that ¤ v; x 0 for all x C and v; c > 0; h i Ä 2 h i which yields v C thus contradicts to the fact that c .C / . 2 ı 2 ı ı Definition 3.15 Let ˝ be a nonempty, convex set. e to ˝ at x ˝ is N 2 T.x ˝/ cl K ˝ x cl RC.˝ x/: NI WD f Ng D N Figure 3.3: Tangent cone and normal cone. Now we get the duality correspondence between normals and tangents to convex sets; see a demonstration in Figure 3.3. eorem 3.16 Let ˝ be a convex set and let x ˝. en we have the equalities N 2 N.x ˝/ T.x ˝/ı and T.x ˝/ N.x ˝/ı: (3.4) NI D NI NI D NI 3.4. TANGENTS TO CONVEX SETS 105 Proof. To verify first the inclusion N.x ˝/ ŒT .x ˝/ , pick any v N.x ˝/ and get by def- NI NI ı 2 NI inition that v; x x 0 for all x ˝. is implies that h Ni Ä 2 ˝v; t.x x/˛ 0 whenever t 0 and x ˝: N Ä 2 Fix w T.x ˝/ and find tk 0 and xk ˝ with tk.xk x/ w as k . en 2 NI 2 N ! ! 1 v; w lim v; tk.xk x/ 0 and thus v T.x ˝/ı: h i D k h N i Ä 2 NI !1 To verify the opposite inclusion in (3.4), fix any v ŒT .x ˝/ and deduce from the relations 2 NI ı v; w 0 as w T.x ˝/ and x x T.x ˝/ as x ˝ that h i Ä 2 NI N 2 NI 2 v; x x 0 for all x ˝: h Ni Ä 2 is yields w N.x ˝/ and thus completes the proof of the first equality in (3.4). e second 2 NI equality in (3.4) follows from the first one due to N.x ˝/ı T.x ˝/ıı cl T.x ˝/ NI D NI D NI by Proposition 3.14 and the closedness of T.x ˝/ by Definition 3.15. NI e next result, partly based on the Farkas lemma from eorem 3.10, provides effective representations of the tangent and normal cones to linear inequality systems. n eorem 3.17 Let ai R , let bi R for i 1; : : : ; m, and let 2 2 D n ˝ ˚x R ˇ ai ; x bi «: WD 2 ˇ h i Ä Suppose that x ˝ and define the active index set I.x/ i 1; : : : ; m ai ; x bi . en N 2 N WD f D j h Ni D g we have the relationships n T.x ˝/ ˚v R ˇ ai ; v 0 for all i I.x/«; (3.5) NI D 2 ˇ h i Ä 2 N N.x ˝/ cone˚ai ˇ i I.x/«: (3.6) NI D ˇ 2 N Here we use the convention that cone 0 . ; WD f g Proof. To justify (3.5), consider the convex cone n K ˚v R ˇ ai ; v 0 for all i I.x/«: WD 2 ˇ h i Ä 2 N It is easy to see that for any x ˝, any i I.x/, and any t 0 we have 2 2 N ˝ai ; t.x x/˛ t ai ; x ai ; x t.bi bi / 0; N D h i h Ni Ä D 106 3. REMARKABLE CONSEQUENCES OF CONVEXITY and hence RC.˝ x/ K. us it follows from the closedness of K that N T.x ˝/ cl RC.˝ x/ K: NI D N To verify the opposite inclusion in (3.5), observe that ai ; x < bi for i I.x/. Using this and h Ni … N picking an arbitrary element v K give us 2 ai ; x tv bi for all i I.x/ and small t > 0: h N C i Ä … N Since ai ; v 0 for all i I.x/, it tells us that ai ; x tv bi as i 1; : : : ; m yielding h i Ä 2 N h N C i Ä D 1 x tv ˝; or v ˝ x T.x ˝/: N C 2 2 t N NI is justifies the tangent cone representation (3.5). Next we prove (3.6) for the normal cone. e first equality in (3.4) ensures that w N.x ˝/ if and only if w; v 0 for all v T.x ˝/: 2 NI h i Ä 2 NI en we see from (3.5) that w N.x ˝/ if and only if 2 NI ai ; v 0 for all i I.x/ w; v 0 h i Ä 2 N H) h i Ä whenever v Rn. e Farkas lemma from eorem 3.10 tells us that w N.x ˝/ if and only if 2 2 NI w cone ai i I.x/ , and thus (3.6) holds. 2 f j 2 N g 3.5 MEAN VALUE THEOREMS e mean value theorem for differentiable functions is one of the central results of classical real analysis. Now we derive its subdifferential counterparts for general convex functions. eorem 3.18 Let f R R be convex, let a < b, and let the interval Œa; b belong to the W ! domain of f , which is an open interval in R. en there is c .a; b/ such that 2 f .b/ f .a/ @f.c/: (3.7) b a 2 Proof. Define the real-valued function g Œa; b R by W ! f .b/ f .a/ g.x/ f .x/ h .x a/ f .a/i WD b a C for which g.a/ g.b/. is implies, by the convexity and hence continuity of g on its open D domain, that g has a local minimum at some c .a; b/. e function 2 f .b/ f .a/ h.x/ h .x a/ f .a/i WD b a C 3.5. MEAN VALUE THEOREMS 107 Figure 3.4: Subdifferential mean value theorem. is obviously differentiable at c, and hence f .b/ f .a/ @h.c/ ˚h0.c/« n o: D D b a e subdifferential Fermat rule and the subdifferential sum rule from Chapter 2 yield f .b/ f .a/ 0 @g.c/ @f.c/ n o; 2 D b a which implies (3.7) and completes the proof of the theorem. To proceed further with deriving a counterpart of the mean value theorem for convex func- tions on Rn (eorem 3.20), we first present the following lemma. Lemma 3.19 Let f Rn R be a convex function and let D be the domain of f , which is an open W ! subset of Rn. Given a; b D with a b, define the function ' R R by 2 ¤ W ! '.t/ f tb .1 t/a; t R : (3.8) WD C 2 en for any number t0 .0; 1/ we have 2 @'.t0/ ˚ v; b a ˇ v @f.c0/« with c0 t0a .1 t0/b: D h i ˇ 2 WD C Proof. Define the vector-valued functions A R Rn and B R Rn by W ! W ! A.t/ .b a/t and B.t/ tb .1 t/a t.b a/ a A.t/ a; t R : WD WD C D C D C 2 108 3. REMARKABLE CONSEQUENCES OF CONVEXITY It is clear that '.t/ .f B/.t/ and that the adjoint mapping to A is A v v; b a , v Rn. D ı D h i 2 By the subdifferential chain rule from eorem 2.51, we have @'.t0/ A@f.c0/ ˚ v; b a ˇ v @f.c0/«; D D h i ˇ 2 which verifies the result claimed in the lemma. eorem 3.20 Let f Rn R be a convex function in the setting of Lemma 3.19. en there W ! exists an element c .a; b/ such that 2 f .b/ f .a/ ˝@f.c/; b a˛ ˚ v; b a ˇ v @f.c/«: 2 WD h i ˇ 2 Proof. Consider the function ' R R from (3.8) for which '.0/ f .a/ and '.1/ f .b/. W ! D D Applying first eorem 3.18 and then Lemma 3.19 to ', we find t0 .0; 1/ such that 2 f .b/ f .a/ '.1/ '.0/ @'.t0/ ˚ v; b a ˇ v @f.c/« D 2 D h i ˇ 2 with c t0a .1 t0/b. is justifies our statement. D C 3.6 HORIZON CONES is section concerns behavior of unbounded convex sets at infinity. Definition 3.21 Given a nonempty, closed, convex subset F of Rn and a point x F , the 2 of F at x is defined by the formula F .x/ ˚d Rn ˇ x td F for all t > 0«: 1 WD 2 ˇ C 2 Note that the horizon cone is also known in the literature as the asymptotic cone of F at the point in question. Another equivalent definition of F .x/ is given by 1 F x F .x/ \ ; 1 D t t>0 which clearly implies that F .x/ is a closed, convex cone in Rn. e next proposition shows that 1 F .x/ is the same for any x F , and so it can be simply denoted by F . 1 2 1 Proposition 3.22 Let F be a nonempty, closed, convex subset of Rn. en we have the equality F .x1/ F .x2/ for any x1; x2 F . 1 D 1 2 3.6. HORIZON CONES 109 Figure 3.5: Horizon cone. Proof. It suffices to verify that F .x1/ F .x2/ whenever x1; x2 F . Taking any direction 1 1 2 d F .x1/ and any number t > 0, we show that x2 td F . Consider the sequence 2 1 C 2 1  1 à xk x1 ktdÁ 1 x2; k N : WD k C C k 2 en xk F for every k because d F .x1/ and F is convex. We also have xk x2 td, and 2 2 1 ! C so x2 td F since F is closed. us d F .x2/. C 2 2 1 Corollary 3.23 Let F be a closed, convex subset of Rn that contains the origin. en F \ tF: 1 D t>0 Proof. It follows from the construction of F and Proposition 3.22 that 1 F 0 1 F F .0/ \ \ F \ tF; 1 D 1 D t D t D t>0 t>0 t>0 which therefore justifies the claim. Proposition 3.24 Let F be a nonempty, closed, convex subset of Rn. For a given element d Rn, 2 the following assertions are equivalent: (i) d F . 2 1 (ii) ere are sequences tk Œ0; / and fk F such that tk 0 and tkfk d. f g 1 f g ! ! 110 3. REMARKABLE CONSEQUENCES OF CONVEXITY Proof. To verify (i) (ii), take d F and fix x F . It follows from the definition that H) 2 1 N 2 x kd F for all k N : N C 2 2 For each k N, this allows us to find fk F such that 2 2 1 1 x kd fk; or equivalently, x d fk: N C D k N C D k 1 Letting tk , we see that tkfk d as k . WD k ! ! 1 To prove (ii) (i), suppose that there are sequences tk Œ0; / and fk F with H) f g 1 f g tk 0 and tkfk d. Fix x F and verify that d F by showing that ! ! 2 2 1 x td F for all t > 0: C 2 Indeed, for any fixed t > 0 we have 0 t tk < 1 when k is sufficiently large. us Ä .1 t tk/x t tkfk x td as k : C ! C ! 1 It follows from the convexity of F that every element .1 t tk/x t tkfk belongs to F . Hence C x td F by the closedness, and thus we get d F . C 2 2 1 Finally in this section, we present the expected characterization of the set boundedness in terms of its horizon cone. eorem 3.25 Let F be a nonempty, closed, convex subset of Rn. en the set F is bounded if and only if its horizon cone is trivial, i.e., F 0 . 1 D f g Proof. Suppose F is bounded and take any d F . By Proposition 3.24, there exist sequences 2 1 tk Œ0; / with tk 0 and fk F such that tkfk d as k . It follows from the f g 1 ! f g ! ! 1 boundedness of F that tkfk 0, which shows that d 0. ! D To prove the converse implication, suppose by contradiction that F is unbounded while F 0 . en there is a sequence xk F with xk . is allows us to form the se- 1 D f g f g k k ! 1 quence of (unit) directions xk dk ; k N : WD xk 2 k k It ensures without loss of generality that dk d as k with d 1. Fix any x F and ! ! 1 k k D 2 observe that for all t > 0 and k N sufficiently large we have 2  t à t uk 1 x xk F WD xk C xk 2 k k k k by the convexity of F . Furthermore, x td F since uk x td and F is closed. us d C 2 ! C 2 F , which is a contradiction that completes the proof of the theorem. 1 3.7. MINIMAL TIME FUNCTIONS AND MINKOWSKI GAUGE 111 3.7 MINIMAL TIME FUNCTIONS AND MINKOWSKI GAUGE e main object of this and next sections is a remarkable class of convex functions known as the minimal time function. ey are relatively new in convex analysis and highly important in applications. Some of their recent applications to location problems are given in Chapter 4. We also discuss relationships between minimal time functions and Minkowski gauge functions well- recognized in convex analysis and optimization. Definition 3.26 Let F be a nonempty, closed, convex subset of Rn and let ˝ Rn be a nonempty set, not necessarily convex. e associated with the sets F and ˝ is defined by T F .x/ inf ˚t 0 ˇ .x tF / ˝ «: (3.9) ˝ WD ˇ C \ ¤ ; Figure 3.6: Minimal time functions. e name comes from the following interpretation: (3.9) signifies the minimal time needed for the point x to reach the target set ˝ along the constant dynamics F ; see Figure 3.6. Note that when F D IB, the closed unit ball in Rn, the minimal time function (3.9) reduces to the distance function d.x ˝/ considered in Section 2.5. In the case where F 0 , the minimal time function (3.9) is I F D f g the indicator function of ˝. If ˝ 0 , we get T .x/ F . x/, where F is the Minkowski D f g ˝ D gauge defined later in this section. e minimal time function (3.9) also covers another interesting class of functions called the directional minimal time function, which corresponds to the case where F is a nonzero vector. In this section we study general properties of the minimal time function (3.9) used in what follows. ese properties are certainly of independent interest. Observe that T F .x/ 0 if x ˝, and that T F .x/ is finite if there exists t 0 with ˝ D 2 ˝ .x tF / ˝ ; C \ ¤ ; which holds, in particular, when 0 int F . If no such t exists, then T F .x/ . us the min- 2 ˝ D 1 imal time function (3.9) is an extended-real-valued function in general. 112 3. REMARKABLE CONSEQUENCES OF CONVEXITY e sets F and ˝ in (3.9) are not necessarily bounded. e next theorem reveals behavior T F of ˝ .x/ in the case where ˝ is bounded, while F is not necessarily bounded. It is an exercise to formulate and prove related results for the case where F is bounded, while ˝ is not necessarily bounded. eorem 3.27 Let F be a nonempty, closed, convex subset of Rn and let ˝ Rn be a nonempty, closed set. Assume that ˝ is bounded. en T F ˝ .x/ 0 implies x ˝ F : D 2 1 e converse holds if we assume additionally that 0 F . 2 F Proof. Suppose that T .x/ 0 and find tk 0, tk 0, such that ˝ D ! .x tkF/ ˝ ; k N : C \ ¤ ; 2 is gives us qk ˝ and fk F with x tkfk qk for each k. By the boundedness and 2 2 C D closedness of ˝, we can select a subsequence of qk (no relabeling) that converges to some f g q ˝, and thus tkfk q x. It follows from Proposition 3.24 that q x F , which justi- 2 ! 2 1 fies x ˝ F . 2 1 To prove the opposite implication, pick any x ˝ F and find q ˝ and d F such 2 1 2 2 1 that x q d. Since 0 F and d F , we get from Corollary 3.23 the inclusions k.q x/ D 2 2 1 D kd F for all k N. is shows that 2 2 1 1 q x F and so x F Á ˝ ; k N : 2 k C k \ ¤ ; 2 1 e latter ensures that 0 T F .x/ for all k N, which holds only when T F .x/ 0 and Ä ˝ Ä k 2 ˝ D thus completes the proof of the theorem. e following example illustrates the result of eorem 3.27. Example 3.28 Let F R Œ 1; 1 R2 and let ˝ be the disk centered at .1; 0/ with radius D r 1. Using the representation of F in Corollary 3.23, we get F R 0 and ˝ F D 1 1 D f g 1 D R Œ 1; 1. us T F .x/ 0 if and only if x R Œ 1; 1. ˝ D 2 e next result characterizes the convexity of the minimal time function (3.9). Proposition 3.29 Let F be a nonempty, closed, convex subset of Rn. If ˝ Rn is nonempty and convex, then the minimal time function (3.9) is convex. If furthermore F is bounded and ˝ is closed, then the converse holds as well. 3.7. MINIMAL TIME FUNCTIONS AND MINKOWSKI GAUGE 113 F Proof. Take any x1; x2 dom T and let .0; 1/. We now show that 2 ˝ 2 F F F T x1 .1 /x2 T .x1/ .1 /T .x2/ (3.10) ˝ C Ä ˝ C ˝ F provided that both sets F and ˝ are convex. Denote i T .xi / for i 1; 2 and, given any WD ˝ D > 0, select numbers ti so that i ti < i and .xi ti F/ ˝ ; i 1; 2: Ä C C \ ¤ ; D It follows from the convexity of F and ˝ that x1 .1 /x2 t1 .1 /t2F ˝ ; C C C \ ¤ ; which implies in turn the estimates F F F T x1 .1 /x2 t1 .1 /t2 T .x1/ .1 /T .x2/ : ˝ C Ä C Ä ˝ C ˝ C T F Since is arbitrary, we arrive at (3.10) and thus verifies the convexity of ˝ . T F Conversely, suppose that ˝ is convex provided that F is bounded and that ˝ is closed. en the level set ˚x Rn ˇ T F .x/ 0« 2 ˇ ˝ Ä is definitely convex. We can easily show set reduces to the target set ˝, which justifies the con- vexity of the latter. Definition 3.30 Let F be a nonempty, closed, convex subset of Rn. e M associated with the set F is defined by n F .x/ inf ˚t 0 ˇ x tF «; x R : (3.11) WD ˇ 2 2 e following theorem summarizes the main properties of Minkowski gauge functions. eorem 3.31 Let F be a closed, convex set that contains the origin. e Minkowski gauge F is an extended-real-valued function, which is positively homogeneous and subadditive. Further- more, F .x/ 0 if and only if x F . D 2 1 Proof. We first prove that F is subadditive meaning that n F .x1 x2/ F .x1/ F .x2/ for all x1; x2 R : (3.12) C Ä C 2 is obviously holds if the right-hand side of (3.12) is equal to infinity, i.e., we have that x1 dom F or x2 dom F . Suppose now that x1; x2 dom F and fix > 0. en there are … … 2 numbers t1; t2 0 such that F .x1/ t1 < F .x1/ ; F .x2/ t2 < F .x2/ and x1 t1F; x2 t2F: Ä C Ä C 2 2 114 3. REMARKABLE CONSEQUENCES OF CONVEXITY It follows from the convexity of F that x1 x2 t1F t2F .t1 t2/F , and so C 2 C C F .x1 x2/ t1 t2 < F .x1/ F .x2/ 2; C Ä C C C which justifies (3.12) since > 0 was chosen arbitrarily. To check the positive homogeneous property of F , we take any ˛ > 0 and conclude that t F .˛x/ inf ˚t 0 ˇ ˛x tF « inf nt 0 ˇ x F o D ˇ 2 D ˇ 2 ˛ t t ˇ n ˇ o ˛ inf 0 ˇ x F ˛F .x/: D ˛ ˇ 2 ˛ D F It remains to verify the last statement of the theorem. Observe that F .x/ T .x/ is a special D ˝ case of the minimal time function with ˝ 0 . Since 0 F , eorem 3.27 tells us that F .x/ D f g 2 D 0 if and only if x ˝ . F/ F , which completes the proof. 2 1 D 1 We say for simplicity that ' Rn R is ` Lipschitz continuous on Rn if it is Lipschitz con- W ! tinuous on Rn with Lipschitz constant ` 0. e next result calculates a constant ` of Lipschitz continuity of the Minkowski gauge on Rn. Proposition 3.32 Let F be a closed, convex set that contains the origin as an interior point. en the n Minkowski gauge F is `-Lipschitz continuous on R with the constant 1 n ˇ o ` inf ˇ IB.0 r/ F; r > 0 : (3.13) WD r ˇ I n In particular, we have F .x/ ` x for all x R . Ä k k 2 Proof. e condition 0 int F ensures that ` < for the constant ` from (3.13). Take any r > 0 2 1 satisfying IB.0 r/ F and get from the definitions that I x F .x/ inf ˚t 0 ˇ x tF « inf ˚t 0 ˇ x tIB.0 r/« inf ˚t 0 x rt« k k; D ˇ 2 Ä ˇ 2 I D j k k Ä D r which implies that F .x/ ` x . en the subadditivity of F gives us that Ä k k n F .x/ F .y/ F .x y/ ` x y for all x; y R ; Ä Ä k k 2 n and thus F is ` Lipschitz continuous on R with constant (3.13). Next we relate the minimal time function to the Minkowski gauge. eorem 3.33 Let F be a nonempty, closed, convex set. en for any ˝ we have ¤ ; F n T .x/ inf ˚F .! x/ ˇ ! ˝«; x R : (3.14) ˝ D ˇ 2 2 3.7. MINIMAL TIME FUNCTIONS AND MINKOWSKI GAUGE 115 Proof. Consider first the case where T F .x/ . In this case we have ˝ D 1 ˚t 0 ˇ .x tF / ˝ « ; ˇ C \ ¤ ; D; which implies that t 0 ! x tF for every ! ˝. us F .! x/ as ! ˝, f j 2 g D ; 2 D 1 2 and hence the right-hand side of (3.14) is also infinity. Considering now the case where T F .x/ < , fix any t 0 such that .x tF / ˝ . ˝ 1 C \ ¤ ; is gives us f F and ! ˝ with x tf !. us ! x tF , and so F .! x/ t, 2 2 C D 2 Ä which yields inf! ˝ F .! x/ t. It follows from definition (3.9) that 2 Ä T F inf F .! x/ ˝ .x/ < : ! ˝ Ä 1 2 To verify the opposite inequality in (3.14), denote inf! ˝ F .! x/ and for any > 0 find WD 2 ! ˝ with F .! x/ < . By definition of the Minkowski gauge (3.11), we get t 0 such 2 C that t < and ! x tF . is gives us ! x tF and therefore C 2 2 C T F .x/ t < : ˝ Ä C Since was chosen arbitrarily, we arrive at T F ˝ .x/ inf F .! x/; Ä D ! ˝ 2 which implies the remaining inequality in (3.14) and completes the proof. As a consequence of the obtained result and Proposition 3.32, we establish now the ` Lipschitz continuity of minimal time function with the constant ` calculated in (3.13). Corollary 3.34 Let F be a closed, convex set that contains the origin as an interior point. en for any nonempty set ˝ the function T F is ` Lipschitz with constant (3.13). ˝ Proof. It follows from the subadditivity of F and the result of Proposition 3.32 that F .! x/ F .! y/ F .y x/ F .! y/ ` y x for all ! ˝ Ä C Ä C k k 2 whenever x; y Rn. en eorem 3.33 tells us that 2 T F .x/ T F .y/ ` x y ; ˝ Ä ˝ C k k which implies the ` Lipschitz property of T F due to the arbitrary choice of x; y. ˝ Define next the enlargement of the target set ˝ via minimal time function by n F ˝r ˚x R ˇ T .x/ r«; r > 0: (3.15) WD 2 ˇ ˝ Ä 116 3. REMARKABLE CONSEQUENCES OF CONVEXITY Proposition 3.35 Let F be a nonempty, closed, convex subset of Rn and let ˝ Rn be a nonempty F set. en for any nonempty set ˝ and any x ˝r with T .x/ < we have … ˝ 1 T F .x/ T F .x/ r: ˝ D ˝r C F F Proof. It follows from ˝ ˝r that T .x/ T .x/ < . Take any t 0 with ˝r Ä ˝ 1 .x tF / ˝r C \ ¤ ; F and find f1 F and u ˝r such that x tf1 u. Since u ˝r , we have T .u/ r. For any 2 2 C D 2 ˝ Ä > 0, there is s 0 with s < r and .u sF / ˝ . is implies the existence of ! ˝ C C \ ¤ ; 2 and f2 F such that u sf2 !. us 2 C D ! u sf2 .x tf1/ sf2 x .t s/F D C D C C 2 C C since F is convex. is gives us T F .x/ t s t r and so T F .x/ t r ˝ Ä C Ä C C ˝ Ä C by the arbitrary choice of > 0. is allows us to conclude that T F .x/ T F r: ˝ Ä ˝r C F To verify the opposite inequality, denote T .x/ and observe that r < by x ˝r . For any WD ˝ … > 0, find t Œ0; /, f F , and ! ˝ with t < and x tf !. Hence 2 1 2 2 Ä C C D ! x tf x .t r/f rf x .t r/f rF and so T F .x .t r/f / r; D C D C C 2 C C ˝ C Ä which shows that x .t r/f ˝r . We also see that x .t r/f x .t r/F . us C 2 C 2 C T F .x/ t r r , which verifies that ˝r Ä Ä C r T F T F .x/ C ˝r Ä D ˝ since > 0 was chosen arbitrarily. is completes the proof. Proposition 3.36 Let F be a nonempty, closed, convex subset of Rn. en for any nonempty set ˝, any x dom T F , any t 0, and any f F we have 2 ˝ 2 T F .x tf / T F .x/ t: ˝ Ä ˝ C Proof. Fix > 0 and select s 0 so that T F .x/ s < T F .x/ and .x sF / ˝ : ˝ Ä ˝ C C \ ¤ ; en .x tf tF sF / ˝ , and hence .x tf .t s/F / ˝ . It shows that C C \ ¤ ; C C \ ¤ ; T F .x tf / t s t T F .x/ ; ˝ Ä C Ä C ˝ C which implies the conclusion of the proposition by letting 0 . ! C 3.8. SUBGRADIENTS OF MINIMAL TIME FUNCTIONS 117 3.8 SUBGRADIENTS OF MINIMAL TIME FUNCTIONS Observing that the minimal time function (3.9) is intrinsically nonsmooth, we study here its gen- eralized differentiation properties in the case where the target set ˝ is convex. In this case (since the constant dynamics F is always assumed convex) function (3.9) is convex as well by Proposi- tion 3.29. e results below present precise formulas for calculating the subdifferential @T F .x/ ˝ N depending on the position of x with respect to the set ˝ F . N 1 eorem 3.37 Let both sets 0 F Rn and ˝ Rn be nonempty, closed, convex. For any 2 T F x ˝ F , the subdifferential of ˝ at x is calculated by N 2 1 N T F @ ˝ .x/ N.x ˝ F / C ; (3.16) N D NI 1 \ n where C v R F . v/ 1 is defined via (2.41). In particular, we have WD f 2 j Ä g F @T .x/ N.x ˝/ C for x ˝: (3.17) ˝ N D NI \ N 2 T F T F Proof. Fix any v @ ˝ .x/. Using (2.13) and the fact that ˝ .x/ 0 for all x ˝ F , we get 2 N D 2 1 v N.x ˝ F /. Write x ˝ F as x ! d with ! ˝ and d F . Fix any f F 2 NI 1 N 2 1 N DN N 2 2 1 2 and define xt .x d/ tf ! tf as t > 0. en .xt tF / ˝ , and so WD N C DN C \ ¤ ; F d v; xt x T .xt / t; or equivalently Dv; f E 1 for all t > 0: h Ni Ä ˝ Ä t Ä By letting t , it implies that v; f 1, and hence we get v C . is verifies the inclu- T F ! 1 h i Ä 2 sion @ ˝ .x/ N.x ˝ F / C . N NI 1 \ To prove the opposite inclusion in (3.16), fix any v N.x ˝ F / C and any u 2 NI 1 \ 2 dom T F . For any > 0, there exist t Œ0; /, ! ˝, and f F such that ˝ 2 1 2 2 T F .u/ t < T F .u/ and u tf !: ˝ Ä ˝ C C D Since ˝ ˝ F , we have the relationships 1 v; u x v; ! x t v; f t < T F .u/ T F .u/ T F .x/ ; h Ni D h Ni C h i Ä ˝ C D ˝ ˝ N C T F T F which show that v @ ˝ .x/. In this way we get N.x ˝ F / C @ ˝ .x/ and thus justify 2 N NI 1 \ N the subdifferential formula (3.16) in the general case where x ˝ F . N 2 1 It remains to verify the simplified representation (3.17) in the case where x ˝. Since N 2 ˝ ˝ F , we obviously have the inclusion 1 N.x ˝ F / C N.x ˝/ C : NI 1 \ NI \ To show the converse inclusion, pick v N.x ˝/ C and fix x ˝ F . Taking ! ˝ 2 NI \ 2 1 2 and d F with x ! d gives us td F since 0 F , so v; td 1 for all t > 0. us 2 1 D 2 2 h i Ä v; d 0, which readily implies that h i Ä v; x x v; ! d x v; ! x v; d 0: h Ni D h Ni D h Ni C h i Ä In this way we arrive at v N.x ˝ F / C and thus complete the proof. 2 NI 1 \ 118 3. REMARKABLE CONSEQUENCES OF CONVEXITY e next theorem provides the subdifferential calculation for minimal time functions in the cases complemented to those in eorem 3.37, i.e., when x ˝ F and x ˝. N … 1 N … eorem 3.38 Let both sets 0 F Rn and ˝ Rn be nonempty, closed, convex and let x T F 2 N 2 dom ˝ for the minimal time function (3.9). Suppose that ˝ is bounded. en for x ˝ F N … 1 we have F @T .x/ N.x ˝r / S ; (3.18) ˝ N D NI \ where the enlargement ˝r is defined in (3.15), and where n F S ˚v R ˇ F . v/ 1« with r T .x/ > 0: WD 2 ˇ D D ˝ N Proof. Pick v @T F .x/. Observe from the definition that 2 ˝ N v; x x T F .x/ T F .x/ for all x Rn : (3.19) h Ni Ä ˝ ˝ N 2 F F en T .x/ r T .x/, and hence v; x x 0 for any x ˝r . is yields v N.x ˝r /. ˝ Ä D ˝ N h Ni Ä 2 2 NI Using (3.19) for x x f with f F and Proposition 3.36 ensures that F . v/ 1. Fix WD N 2 Ä .0; r/ and select t R, f F , and ! ˝ so that 2 2 2 2 r t < r 2 and ! x tf: Ä C DN C We can write ! x f .t /f and then see that T F .x f / t . Applying (3.19) DN C C ˝ N C Ä to x x f gives us the estimates WD N C v; f T F .x f / T F .x/ t r 2 : h i Ä ˝ N C ˝ N Ä Ä which imply in turn that 1 v; f F . v/: Ä h i Ä Letting now 0 tells us that F . v/ 1, i.e., F . v/ 1 in this case. us v S , which # F D 2 proves the inclusion @T .x/ N.x ˝r / S . ˝ N NI \ To verify the opposite inclusion in (3.18), take any v N.x ˝r / with F . v/ 1 and 2 NI D show that (3.19) holds. It follows from eorem 3.37 that v @T F .x/, and so 2 ˝r N v; x x T F .x/ for all x Rn : h Ni Ä ˝r 2 Fix x Rn and deduce from Proposition 3.35 that T F .x/ r T F .x/ in the case t 2 ˝ D ˝r WD T F .x/ > r, which ensures (3.19). Suppose now that t r and for any > 0 choose f F ˝ Ä 2 such that v; f > 1 . en Proposition 3.36 tells us that T F .x .r t/f / r, and so h i ˝ Ä x .r t/f ˝r . Since v N.x ˝r /, one has v; x .r t/f x 0, which yields 2 2 NI h Ni Ä v; x x v; f .r t/ .1 /.t r/ .1 / T F .x/ T F .x/: h Ni Ä h i Ä D ˝ ˝ N Since > 0 was arbitrarily chosen, we get (3.19), and hence verify that v @T F .x/. 2 ˝ N 3.8. SUBGRADIENTS OF MINIMAL TIME FUNCTIONS 119 e next result addresses the same setting as in eorem 3.38 while presenting another formula for calculating the subdifferential of the minimal time function that involves the subdif- ferential of the Minkowski gauge and generalized projections to the target set. eorem 3.39 Let both sets 0 F Rn and ˝ Rn be nonempty, closed, convex and let T F 2 x dom ˝ . Suppose that ˝ is bounded and that x ˝ F . en we have N 2 N … 1 F @T .x/ @F .! x/ N.! ˝/ (3.20) ˝ N D N N \ N I for any ! ˘F .x ˝/ via the generalized projector N 2 NI F ˘F .x ˝/ ˚! ˝ ˇ T .x/ F .! x/«: (3.21) NI WD 2 ˇ ˝ N D N F F Proof. Fixing any v @T .x/ and ! ˘F .x ˝/, we get T .x/ F .! x/ and 2 ˝ N N 2 NI ˝ N D N N v; x x T F .x/ T F .x/ for all x Rn : (3.22) h Ni Ä ˝ ˝ N 2 It is easy to check the following relationships that hold for any ! ˝: 2 v; ! ! v; .! ! x/ x T F .! ! x/ T F .x/ h N i D h N CN Ni Ä ˝ N CN ˝ N F ! .! ! x/ F .! x/ 0: Ä N CN N N D is shows that v N.! ˝/. Denote now u ! x and for any t .0; 1/ and u Rn get 2 N I e WD N N 2 2 T F T F v; t.u u/ ˝ x t.u u/ ˝ .x/ h e i Ä N e N F ! .x t.u u// F .! x/ Ä N N e N N F .u t.u u// F .u/ D e C e e F tu .1 t/u F .u/ D C e e tF .u/ .1 t/F .u/ F .u/ Ä C e e t F .u/ F .u/ D e by applying (3.22) with x x t.u u/. It follows that WD N e n v; u u F .u/ F .u/ for all u R : h ei Ä e 2 us v @F .u/, and so v @F .! x/, which proves the inclusion “ ” in (3.20). 2 2 N N To verify thee opposite inclusion, write F F .x/ inf ˚t 0 ˇ x tF « inf ˚t 0 ˇ . x tF / O « T . x/ D ˇ 2 D ˇ C \ ¤ ; D O with O 0 . Since x ˝ F , for any ! ˝ we have x ! O F F . Further- WD f g N … 1 N 2 N N … 1 D 1 more, it follows from v @F .! x/ by applying eorem 3.38 that v S . Hence it remains 2 N N 2 to justify the inclusion @F .! x/ N.! ˝/ N.x ˝r / (3.23) N N \ N I NI 120 3. REMARKABLE CONSEQUENCES OF CONVEXITY and then to apply eorem 3.38. To proceed with the proof of (3.23), pick any vector x ˝r 2 and a positive number arbitrarily small. en there are t < r , f F , and ! ˝ such that C 2 2 ! x tf , which yields D C v; x x v; ! tf x h Ni D h Ni t v; f v; ! ! v; ! x D h i C h N i C h N Ni t v; ! ! v; ! x Ä C h N i C h N Ni T F .x/ v; ! ! v; ! x : Ä ˝ N C C h N i C h N Ni Observe that v; ! ! 0 by v N.! ˝/. Moreover, h N i Ä 2 N I F v; ! x v; 0 .! x/ F .0/ F .! x/ T .x/ h N Ni D h N N i Ä N N D ˝ N since v @F .! x/. It follows that v; x x for all x ˝r . Hence v N.x ˝r / be- 2 N N h Ni Ä 2 2 NI cause > 0 was chosen arbitrarily. is completes the proof of the theorem. 3.9 NASH EQUILIBRIUM is section gives a brief introduction to noncooperative game theory and presents a simple proof of the existence of Nash equilibrium as a consequence of convexity and Brouwer’s fixed point theorem. For simplicity, we only consider two-person games while more general situations can be treated similarly. n p Definition 3.40 Let ˝1 and ˝2 be nonempty subsets of R and R , respectively. A - in the case of two players I and II consists of two strategy sets ˝i and two real-valued functions ui ˝1 ˝2 R for i 1; 2 called the . en we refer to this game W ! D as ˝i ; ui for i 1; 2. f g D e key notion of Nash equilibrium was introduced by John Forbes Nash, Jr., in 1950 who proved the existence of such an equilibrium and was awarded the Nobel Memorial Prize in Eco- nomics in 1994. Definition 3.41 Given a noncooperative two-person game ˝i ; ui , i 1; 2, a N - f g D is an element .x1; x2/ ˝1 ˝2 satisfying the conditions N N 2 u1.x1; x2/ u1.x1; x2/ for all x1 ˝1; N Ä N N 2 u2.x1; x2/ u2.x1; x2/ for all x2 ˝2: N Ä N N 2 e conditions above mean that .x1; x2/ is a pair of best strategies that both players agree N N with in the sense that x1 is a best response of Player I when Player II chooses the strategy x2, and N N x2 is a best response of Player II when Player I chooses the strategy x1. Let us now consider two N N examples to illustrate this concept. 3.9. NASH EQUILIBRIUM 121 Example 3.42 In a two-person game suppose that each player can only choose either strategy A or B. If both players choose strategy A, they both get 4 points. If Player I chooses strategy A and Player II chooses strategy B, then Player I gets 1 point and Player II gets 3 points. If Player I chooses strategy B and Player II chooses strategy A, then Player I gets 3 points and Player II gets 1 point. If both choose strategy B, each player gets 2 points. e payoff function of each player is represented in the payoff matrix below. Player II A B A 4;. 4 1;. 3 . 3;. 1 2;. 2 Player I B In this example, Nash equilibrium occurs when both players choose strategy A, and it also occurs when both players choose strategy B. Let us consider, for instance, the case where both players choose strategy B. In this case, given that Player II chooses strategy B, Player I also wants to keep strategy B because a change of strategy would lead to a reduction of his/her payoff from 2 to 1. Similarly, given that Player I chooses strategy B, Player II wants to keep strategy B because a change of strategy would also lead to a reduction of his/her payoff from 2 to 1. Example 3.43 Let us consider another simple two-person game called matching pennies. Sup- pose that Player I and Player II each has a penny. Each player must secretly turn the penny to heads or tails and then reveal his/her choices simultaneously. Player I wins the game and gets Player II’s penny if both coins show the same face (heads-heads or tails-tails). In the other case where the coins show different faces (heads-tails or tails-heads), Player II wins the game and gets Player I’s penny. e payoff function of each player is represented in the payoff matrix below. Player II Heads Tails Heads 1; . 1 1;. 1 . 1;. 1 1; . 1 Player I Tails In this game it is not hard to see that no Nash equilibrium exists. Denote by A the matrix that represents the payoff function of Player I, and denote by B the matrix that represents the payoff function of Player II: 1 1 1 1 A and B D 1 1 D 1 1 122 3. REMARKABLE CONSEQUENCES OF CONVEXITY Now suppose that Player I is playing with a mind reader who knows Player I’s choice of faces. If Player I decides to turn heads, then Player II knows about it and chooses to turn tails and wins the game. In the case where Player I decides to turn tails, Player II chooses to turn heads and again wins the game. us to have a fair game, he/she decides to randomize his/her strategy by, e.g., tossing the coin instead of putting the coin down. We describe the new game as follows. Player II Heads, q1 Tails, q2 Heads, p1 1; . 1 1;. 1 . p 1;. 1 1; . 1 Player I Tails, 2 In this new game, Player I uses a coin randomly with probability of coming up heads p1 and prob- ability of coming up tails p2, where p1 p2 1. Similarly, Player II uses another coin randomly C D with probability of coming up heads q1 and probability of coming up tails q2, where q1 q2 1. C D e new strategies are called mixed strategies while the original ones are called pure strategies. Suppose that Player II uses mixed strategy q1; q2 . en Player I’s expected payoff for f g playing the pure strategy heads is uH .q1; q2/ q1 q2 2q1 1: D D Similarly, Player I’s expected payoff for playing the pure strategy tails is uT .q1; q2/ q1 q2 2q1 1: D C D C us Player I’s expected payoff for playing mixed strategy p1; p2 is f g T u1.p; q/ p1uH .q1; q2/ p2uT .q1; q2/ p1.q1 q2/ p2. q1 q2/ p Aq; D C D C C D T T where p Œp1; p2 and q Œq1; q2 . D D By the same arguments, if Player I chooses mixed strategy p1; p2 , then Player II’s ex- f g pected payoff for playing mixed strategy q1; q2 is f g T u2.p; q/ p Bq u1.p; q/: D D For this new game, an element .p; q/ is a Nash equilibrium if N N 2 u1.p; q/ u1.p; q/ for all p ; N Ä N N 2 u2.p; q/ u2.p; q/ for all q ; N Ä N N 2 where ˚.p1; p2/ ˇ p1 0; p2 0; p1 p2 1« is a nonempty, compact, convex set. WD ˇ C D Nash [23, 24] proved the existence of his equilibrium in the class of mixed strategies in a more general setting. His proof was based on Brouwer’s fixed point theorem with rather involved 3.9. NASH EQUILIBRIUM 123 arguments. Now we present, following Rockafellar [28], a much simpler proof of the Nash equi- librium theorem that also uses Brouwer’s fixed point theorem while applying in addition just elementary tools of convex analysis and optimization. n p eorem 3.44 Consider a two-person game ˝i ; ui , where ˝1 R and ˝2 R are f g nonempty, compact, convex sets. Let the payoff functions ui ˝1 ˝2 R be given by W ! T T u1.p; q/ p Aq; u2.p; q/ p Bq; WD WD where A and B are n p matrices. en this game admits a Nash equilibrium. Proof. It follows from the definition that an element .p; q/ ˝1 ˝2 is a Nash equilibrium of N N 2 the game under consideration if and only if u1.p; q/ u1.p; q/ for all p ˝1; N N N 2 u2.p; q/ u2.p; q/ for all q ˝2: N N N 2 It is a simple exercise to prove (see also an elementary convex optimization result of Corollary 4.15 in Chapter 4) that this holds if and only if we have the normal cone inclusions pu1.p; q/ N.p ˝1/; qu2.p; q/ N.q ˝2/: (3.24) r N N 2 NI r N N 2 NI By using the structures of ui and defining ˝ ˝1 ˝2, conditions (3.24) can be expressed in WD one of the following two equivalent ways: T Aq N.p ˝1/ and B p N.q ˝2/ N 2 NI N 2 NI I T .Aq; B p/ N.p ˝1/ N.q ˝2/ N..p; q/ ˝1 ˝2/ N..p; q/ ˝/: N N 2 NI NI D N N I D N N I By Proposition 1.78 on the Euclidean projection, this is equivalent also to .p; q/ ˘..p; q/ .Aq; BT p/ ˝/: (3.25) N N D N N C N N I Defining now the mapping F ˝ ˝ by W ! T F .p; q/ ˘ .p; q/ .Aq; B p/ ˝/ with ˝ ˝1 ˝2 WD C I D and employing another elementary projection property from Proposition 1.79 allow us to con- clude that the mapping F is continuous and the set ˝ is nonempty, compact, and convex. By the classical Brouwer fixed point theorem (see, e.g., [33]), the mapping F has a fixed point .p; q/ ˝, N N 2 which satisfies (3.25), and thus it is a Nash equilibrium of the game. 124 3. REMARKABLE CONSEQUENCES OF CONVEXITY 3.10 EXERCISES FOR CHAPTER 3 Exercise 3.1 Let f Rn R be a function finite around x. Show that f is Gâteaux dif- W ! N ferentiable at x with f .x/ v if and only if the directional derivative f .x d/ exists and N G0 N D 0 NI f .x d/ v; d for all d Rn. 0 NI D h i 2 Exercise 3.2 Let ˝ be a nonempty, closed, convex subset of Rn. (i) Prove that the function f .x/ Œd.x ˝/2 is differentiable on Rn and WD I f .x/ 2Œx ˘.x ˝/ r N D N NI for every x Rn. N 2 (ii) Prove that the function g.x/ d.x ˝/ is differentiable on Rn ˝. WD I n Exercise 3.3 Let F be a nonempty, compact, convex subset of Rn and let A Rn Rp be a W ! linear mapping. Define 1 f .x/ sup Ax; u u 2: WD u F h i 2k k 2 Prove that f is convex and differentiable. Exercise 3.4 Give an example of a function f Rn R, which is Gâteuax differentiable but W ! not Fréchet differentiable. Can such a function be convex? @f Exercise 3.5 Let f Rn R be a convex function and let x int.domf /. Show that if W ! N 2 @xi at x exists for all i 1; : : : ; n, then f is (Fréchet) differentiable at x. N D N Exercise 3.6 Let f Rn R be differentiable on Rn. Show that f is strictly convex if and W ! only if it satisfies the condition f .x/; x x < f .x/ f .x/ for all x Rn and x Rn; x x: hr N Ni N 2 N 2 ¤ N Exercise 3.7 Let f Rn R be twice continuously differentiable (C 2) on Rn. Prove that if W ! the Hessian 2f .x/ is positive definite for all x Rn, then f is strictly convex. Give an example r N N 2 showing that the converse is not true in general. Exercise 3.8 A convex function f Rn R is called strongly convex with modulus c > 0 if n W ! for all x1; x2 R and .0; 1/ we have 2 2 1 2 f x1 .1 /x2 f.x1/ .1 /f .x2/ c.1 / x1 x2 : C Ä C 2 k k 3.10. EXERCISES FOR CHAPTER 3 125 Prove that the following are equivalent for any C 2 convex function f Rn R: W ! (i) f strongly convex. c (ii) f 2 is convex. 2k k (iii) 2f .x/d; d c d 2 for all x; d Rn. hr i k k 2 Exercise 3.9 A set-valued mapping F Rn Rn is said to be monotone if W ! v2 v1; x2 x1 0 whenever vi F .xi /; i 1; 2: h i 2 D F is called strictly monotone if v2 v1; x2 x1 > 0 whenever vi F .xi /; x1 x2; i 1; 2: h i 2 ¤ D It is called strongly monotone with modulus ` > 0 if 2 v2 v1; x2 x1 ` x1 x2 whenever vi F.xi /; i 1; 2: h i k k 2 D (i) Prove that for any convex function f Rn R the mapping F.x/ @f.x/ is monotone. W ! WD (ii) Furthermore, for any convex function f Rn R the subdifferential mapping @f.x/ is W ! strictly monotone if and only if f is strictly convex, and it is strongly monotone if and only if f is strongly convex. Exercise 3.10 Prove Corollary 3.7. Exercise 3.11 Let A be an p n matrix and let b Rp. Use the Farkas lemma to show that the 2 system Ax b, x 0 has a feasible solution x Rn if and only if the system AT y 0, bT y > 0 D 2 Ä has no feasible solution y Rp. 2 Exercise 3.12 Deduce Carathéodory’s theorem from Radon’s theorem and its proof. Exercise 3.13 Use Carathéodory’s theorem to prove Helly’s theorem. Exercise 3.14 Let C be a subspace of Rn. Show that the polar of C is represented by n C ı C ? ˚v R ˇ v; c 0 for all c C «: D WD 2 ˇ h i D 2 Exercise 3.15 Let C be a nonempty, convex cone of Rn. Show that n C ı ˚v R ˇ v; c 1 for all c C «: D 2 ˇ h i Ä 2 Exercise 3.16 Let ˝1 and ˝2 be convex sets with int ˝1 ˝2 . Show that \ ¤ ; T.x ˝1 ˝2/ T.x ˝1/ T.x ˝2/ for any x ˝1 ˝2: NI \ D NI \ NI N 2 \ 126 3. REMARKABLE CONSEQUENCES OF CONVEXITY Exercise 3.17 Let A be p n matrix, let ˝ ˚x Rn ˇ Ax 0; x 0«; WD 2 ˇ D and let x .x1;:::; xn/ ˝. Verify the tangent and normal cone formulas: N D N N 2 n T.x ˝/ u .u1; : : : ; un/ R ˇ Au 0; ui 0 for any i with xi 0«; NI D f D 2 ˇ D N D n p n N.x ˝/ ˚v R ˇ v Ay z; y R ; z R ; zi 0; z; x 0«: NI D 2 ˇ D 2 2 h Ni D Exercise 3.18 Give an example of a closed, convex set ˝ R2 with a point x ˝ such that N 2 the cone RC.˝ x/ is not closed. N Exercise 3.19 Let f Rn R be a convex function such that there is ` 0 satisfying @f.x/ W ! `IB. Show that f is ` Lipschitz continuous on Rn. Exercise 3.20 Let f R R be convex. Prove that f is nondecreasing if and only if @f.x/ W ! Œ0; / for all x R. 1 2 Exercise 3.21 Find the horizon cones of the following sets: (i) F .x; y/ R2 y x2 . WD f 2 j g (ii) F .x; y/ R2 y x . WD f 2 j j jg (iii) F .x; y/ R2 x > 0; y 1=x . WD f 2 j g (iv) F ˚.x; y/ R2 x > 0; y px2 1 . WD 2 j C g n Exercise 3.22 Let vi R and bi > 0 for i 1; : : : ; m. Consider the set 2 D n F ˚x R ˇ vi ; x bi for all i 1; : : : ; m : (3.26) WD 2 ˇ h i Ä D g Prove the following representation of the horizon cone and Minkowski gauge for F : n F ˚x R ˇ vi ; x 0 for all i 1; : : : ; m«; 1 D 2 ˇ h i Ä D vi ; u F .u/ max n0; max h io: D i 1;:::;m bi 2f g Exercise 3.23 Let A; B be nonempty, closed, convex subsets of Rn. (i) Show that .A B/ A B . 1 D 1 1 (ii) Find conditions ensuring that .A B/ A B . C 1 D 1 C 1 3.10. EXERCISES FOR CHAPTER 3 127 Exercise 3.24 Let F be a closed, bounded, convex set and let ˝ be a closed set in ¤ ; ¤ ; Rn. Prove that T F .x/ 0 if and only if x ˝. is means that the condition 0 F in eo- ˝ D 2 2 rem 3.27(i) is not required in this case. Exercise 3.25 Let F be a closed, bounded, convex set and let ˝ be a closed, convex ¤ ; ¤ ; set. Consider the sets C and S defined in eorem 3.37 and eorem 3.38, respectively. Prove the following: (i) If x ˝, then N 2 F @T .x/ N.x ˝/ C : ˝ N D NI \ (ii) If x ˝, then N … F F @T .x/ N.x ˝r / S ; r T .x/ > 0: ˝ N D NI \ WD ˝ N (iii) If x ˝, then N … F @T .x/ @F .! x/ N.! ˝/ ˝ N D N N \ N I for any ! ˘F .x ˝/ N 2 NI Exercise 3.26 Give a direct proof of characterizations (3.24) of Nash equilibrium. Exercise 3.27 In Example 3.43 with mixed strategies, find .p; q/ that yields a Nash equilibrium N N in this game. 129 C H A P T E R 4 Applications to Optimization and Location Problems In this chapter we give some applications of the results of convex analysis from Chapters 1–3 to selected problems of convex optimization and facility location. Applications to optimization problems are mainly classical, being presented however in a simplified and unified way. On the other hand, applications to location problems are mostly based on recent journal publications and have not been systematically discussed from the viewpoint of convex analysis in monographs and textbooks. 4.1 LOWER SEMICONTINUITY AND EXISTENCE OF MINIMIZERS We first discuss the notion of lower semicontinuity for extended-real-valued functions, which plays a crucial role in justifying the existence of absolute minimizers and related topics. Note that most of the results presented below do not require the convexity of f . Definition 4.1 An extended-real-valued function f Rn R is .l.s.c./ W ! x Rn if for every ˛ R with f .x/ > ˛ there is ı > 0 such that N 2 2 N f .x/ > ˛ for all x IB.x ı/: (4.1) 2 NI We simply say that f is if it is l.s.c. at every point of Rn. e following examples illustrate the validity or violation of lower semicontinuity. Example 4.2 (i) Define the function f R R by W ! (x2 if x 0; f .x/ ¤ WD 1 if x 0: D It is easy to see that it is l.s.c. at x 0 and in fact on the whole real line. On the other hand, the N D replacement of f .0/ 1 by f .0/ 1 destroys the lower semicontinuity at x 0. D D N D 130 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Figure 4.1: Which function is l.s.c? (ii) Consider the indicator function of the set Œ 1; 1 R given by (0 if x 1; f .x/ j j Ä WD otherwise: 1 en f is a lower semicontinuous function on R. However, the small modification (0 if x < 1; g.x/ j j WD otherwise 1 violates the lower semicontinuity on R. In fact, g is not l.s.c. at x 1 while it is lower semi- D ˙ continuous at any other point of the real line. e next four propositions present useful characterizations of lower semicontinuity. Proposition 4.3 Let f Rn R and let x dom f . en f is l.s.c. at x if and only if for any W ! N 2 N > 0 there is ı > 0 such that f .x/ < f .x/ whenever x IB.x ı/: (4.2) N 2 NI Proof. Suppose that f is l.s.c. at x. Since f .x/ < f .x/, there is ı > 0 with N WD N N f .x/ < f .x/ for all x IB.x ı/: N D 2 NI Conversely, suppose that for any > 0 there is ı > 0 such that (4.2) holds. Fix any < f .x/ and N choose > 0 for which < f .x/ . en N < f .x/ < f .x/ whenever x IB.x ı/ N 2 NI for some ı > 0, which completes the proof. 4.1. LOWER SEMICONTINUITY AND EXISTENCE OF MINIMIZERS 131 Proposition 4.4 e function f Rn R is l.s.c. if and only if for any number R the level set W ! 2 x Rn f .x/ is closed. f 2 j Ä g Proof. Suppose that f is l.s.c. For any fix R, denote 2 ˝ ˚x Rn ˇ f .x/ «: WD 2 ˇ Ä If x ˝, we have f .x/ > . By definition, find ı > 0 such that N … N f .x/ > for all x IB.x ı/; 2 NI which shows that IB.x ı/ ˝c, the complement of ˝. us ˝c is open, and so ˝ is closed. NI To verify the converse implication, suppose that the set x Rn f .x/ is closed for f 2 j Ä g any R. Fix x Rn and R with f .x/ > and then let 2 N 2 2 N ˚x Rn ˇ f .x/ «: WD 2 ˇ Ä Since is closed and x , there is ı > 0 such that IB.x ı/ c. is implies N … NI f .x/ > for all x IB.x ı/ 2 NI and thus completes the proof of the proposition. Proposition 4.5 e function f Rn R is l.s.c. if and only if its epigraph is closed. W ! Proof. To verify the “only if ” part, fix .x; / epi f meaning < f .x/. For any > 0 with N … N f .x/ > > , there is ı > 0 such that f .x/ > whenever x IB.x ı/. us N C C 2 NI IB.x ı/ . ; / .epi f /c; NI C and hence the set epi f is closed. Conversely, suppose that epi f is closed. Employing Proposition 4.4, it suffices to verify that for any R the set ˝ x Rn f .x/ is closed. To see this, fix any sequence 2 WD f 2 j Ä g xk ˝ that converges to x. Since f .xk/ , we have .xk; / epi f for all k. It follows from f g N Ä 2 .xk; / .x; / that .x; / epi f , and so f .x/ . us x ˝, which justifies the closedness ! N N 2 N Ä N 2 of the set ˝. n Proposition 4.6 e function f R R is l.s.c. at x if and only if for any sequence xk x we W ! N !N have lim infk f .xk/ f .x/. !1 N 132 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Proof. If f is l.s.c. at x, take a sequence of xk x. For any < f .x/, there exists ı > 0 N !N N such that (4.1) holds. en xk IB.x ı/, and so < f .xk/ for large k. It follows that 2 NI Ä lim infk f .xk/, and hence f .x/ lim infk f .xk/ by the arbitrary choice of < f .x/. !1 N Ä !1 N To verify the converse, suppose by contradiction that f is not l.s.c. at x. en there is < N 1 f .x/ such that for every ı > 0 we have x IB.x ı/ with f .x /. Applying this to ı ı ı k k N n 2 NI WD gives us a sequence xk R converging to x with f .xk/. It yields f g N f .x/ > lim inf f .xk/; N k !1 which is a contradiction that completes the proof. Now we are ready to obtain a significant result on the lower semicontinuity of the minimal value function (3.9), which is useful in what follows. eorem 4.7 In the setting of eorem 3.27 with 0 F we have that the minimal time function T F 2 ˝ is lower semicontinuous. n Proof. Pick x R and fix an arbitrary sequence xk converging to x as k . Based on N 2 f g N ! 1 Proposition 4.6, we check that T F T F lim inf ˝ .xk/ ˝ .x/: k N !1 T F e above inequality obviously holds if lim infk ˝ .xk/ ; thus it remains to consider the T F !1 D 1 T F case of lim infk ˝ .xk/ Œ0; /. Since it suffices to show that ˝ .x/, we assume !1 D F2 1 N without loss of generality that T .xk/ . By the definition of the minimal time function, we ˝ ! find a sequence tk Œ0; / such that f g 1 F F 1 T .xk/ tk < T .xk/ and .xk tkF/ ˝ as k N : ˝ Ä ˝ C k C \ ¤ ; 2 For each k N, fix fk F and !k ˝ with xk tkfk !k and suppose without loss of gen- 2 2 2 C D erality that !k ! ˝. Consider the two cases: > 0 and 0. If > 0, then ! 2 D ! x fk N F as k ; ! 2 ! 1 which implies that .x F / ˝ , and hence T F .x/. Consider the other case where N C \ ¤ ; ˝ N 0. In this case the sequence tk converges to 0, and so tkfk ! x. It follows from Propo- D f g ! N sition 3.24 that ! x F , which yields x ˝ F . Employing eorem 3.27 tells us that N 2 1 N 2 1 T F .x/ 0 and also T F .x/. is verifies the lower semicontinuity of T F in setting of e- ˝ N D ˝ N ˝ orem 3.27. 4.1. LOWER SEMICONTINUITY AND EXISTENCE OF MINIMIZERS 133 e following two propositions indicate two cases of preserving lower semicontinuity under important operations on functions. n m Proposition 4.8 Let fi R R, i 1; : : : ; m, be l.s.c. at some point x. en the sum P fi W ! D N i 1 is l.s.c. at this point. D Proof. It suffices to verify it for m 2. Taking xk x, we get D !N lim inf fi .xk/ f .x/ for i 1; 2; k N D !1 which immediately implies that lim inf f1.xk/ f2.xk/ lim inf f1.xk/ lim inf f2.xk/ f1.x/ f2.x/; k C k C k N C N !1 !1 !1 and thus f1 f2 is l.s.c. at x by Proposition 4.6. C N n Proposition 4.9 Let I be any nonempty index set and let fi R R be l.s.c. for all i I . en W ! 2 the supremum function f .x/ supi I fi .x/ is also lower semicontinuous. WD 2 Proof. For any number R, we have directly from the definition that 2 n n n ˚x R ˇ f .x/ « ˚x R ˇ sup fi .x/ « \ ˚x R ˇ fi .x/ «: 2 ˇ Ä D 2 ˇ Ä D 2 ˇ Ä i I i I 2 2 us all the level sets of f are closed as intersections of closed sets. is insures the lower semi- continuity of f by Proposition 4.4. e next result provides two interrelated unilateral versions of the classical Weierstrass ex- istence theorem presented in eorem 1.18. e crucial difference is that now we assume the lower semicontinuity versus continuity while ensuring the existence of only minima, not together with maxima as in eorem 1.18. eorem 4.10 Let f Rn R be a l.s.c. function. e following hold: W ! (i) e function f attains its absolute minimum on any nonempty, compact subset ˝ Rn that intersects its domain. (ii) Assume that inf f .x/ x Rn < and that there is a real number > inf f .x/ x Rn f j 2 g 1 f j 2 g for which the level set x Rn f .x/ < of f is bounded. en f attains its absolute minimum f 2 j g on Rn at some point x dom f . N 2 134 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Proof. To verify (i), show first that f is bounded below on ˝. Suppose on the contrary that for every k N there is xk ˝ such that f .xk/ k. Since ˝ is compact, we get without loss of 2 2 Ä generality that xk x ˝. It follows from the lower semicontinuity that !N 2 f .x/ lim inf f .xk/ ; N Ä k D 1 !1 which is a contradiction showing that f is bounded below on ˝. To justify now assertion (i), observe that infx ˝ f .x/ < . Take xk ˝ such that f .xk/ . By the compactness WD 2 1 f g ! of ˝, suppose that xk x ˝ as k . en !N 2 ! 1 f .x/ lim inf f .xk/ ; N Ä k D !1 which shows that f .x/ , and so x realizes the absolute minimum of f on ˝. N D N It remains to prove assertion (ii). It is not hard to show that f is bounded below, so n WD infx Rn f .x/ is a real number. Let xk R for which f .xk/ as k and let k0 N 2 f g !n ! 1 2 be such that f .xk/ < for all k k0. Since the level set x R f .x/ < is bounded, we f 2 j g select a subsequence of xk (no relabeling) that converges to some x. en f g N f .x/ lim inf f .xk/ ; N Ä k Ä !1 which shows that x dom f is a minimum point of f on Rn. N 2 In contrast to all the results given above in this section, the next theorem requires convexity. It provides effective characterizations of the level set boundedness imposed in eorem 4.10(ii). Note that condition (iv) of the theorem below is known as coercivity. eorem 4.11 Let f Rn R be convex and let inf f .x/ x Rn < . e following as- W ! f j 2 g 1 sertions are equivalent: n (i) ere exists a real number ˇ > inf f .x/ x R for which the level set Lˇ x f j 2 g WD f 2 Rn f .x/ < ˇ is bounded. j g (ii) All the level sets x Rn f .x/ < of f are bounded. f 2 j g (iii) lim x f .x/ . k k!1 f .x/D 1 (iv) lim inf x > 0. k k!1 x k k Proof. Let inf f .x/ x Rn , which is a real number or . We first verify (i) (ii). WD f j 2 g 1 H) Suppose by contradiction that there is R such that the level set L is not bounded. en 2 there exists a sequence xk L with xk . Fix ˛ . ; ˇ/, x L˛ and define the convex f g k k ! 1 2 N 2 combination 1 1 1 yk x xk xÁ xk 1 Áx: WD N C p xk N D p xk C p xk N k k k k k k 4.2. OPTIMALITY CONDITIONS 135 By the convexity of f , we have the relationships 1 1 1 f .yk/ f .xk/ .1 /f .x/ < .1 /˛ ˛ < ˇ; Ä p xk C p xk N p xk C p xk ! k k k k k k k k and hence yk Lˇ for k sufficiently large, which ensures the boundedness of yk . It follows 2 f g from the triangle inequality that x 1 1 k Á Á yk 1 x p xk 1 x ; k k p xk p xk k Nk k k p xk k Nk ! 1 k k k k k k which is a contradiction. us (ii) holds. Next we show that (ii) (iii). Indeed, the violation of (iii) yields the existence of a constant H) M > 0 and a sequence xk such that xk as k and f .xk/ < M for all k. But this f g k k ! 1 ! 1 obviously contradicts the boundedness of the level set x Rn f .x/ < M . f 2 j g Finally, we justify (iii) (i) leaving the proof of (iii) (iv) as an exercise. Suppose that H) ” (iii) is satisfied and for M > , find ˛ such that f .x/ M whenever x > ˛. en k k n LM ˚x R ˇ f .x/ < M IB.0 ˛/; WD 2 ˇ g I which shows that this set is bounded and thus verifies (i). 4.2 OPTIMALITY CONDITIONS In this section we first derive necessary and sufficient optimality conditions for optimal solutions to the following set-constrained convex optimization problem: minimize f .x/ subject to x ˝; (4.3) 2 where ˝ is a nonempty, convex set and f Rn R is a convex function. W ! Definition 4.12 We say that x ˝ is a to (4.3) if f .x/ < and there N 2 N 1 is > 0 such that f .x/ f .x/ for all x IB.x / ˝: (4.4) N 2 NI \ If f .x/ f .x/ for all x ˝, then x ˝ is a to (4.3). N 2 N 2 e next result shows, in particular, that there is no difference between local and global solutions for problems of convex optimization. Proposition 4.13 e following are equivalent in the convex setting of (4.3): (i) x is a local optimal solution to the constrained problem (4.3). N (ii) x is a local optimal solution to the unconstrained problem N minimize g.x/ f .x/ ı.x ˝/ on Rn : (4.5) WD C I 136 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS (iii) x is a global optimal solution to the unconstrained problem (4.5). N (iv) x is a global optimal solution to the constrained problem (4.3). N Proof. To verify (i) (ii), select > 0 such that (4.4) is satisfied. Since g.x/ f .x/ for x H) D 2 ˝ IB.x / and since g.x/ for x IB.x / and x ˝, we have \ NI D 1 2 NI … g.x/ g.x/ for all x IB.x /; N 2 NI which means (ii). Implication (ii) (iii) is a consequence of Proposition 2.33. e verification H) of (iii) (iv) is similar to that of (i) (ii), and (vi) (i) is obvious. H) H) H) In what follows we use the term “optimal solutions” for problems of convex optimization, without mentioning global or local. Observe that the objective function g of the unconstrained problem (4.5) associated with the constrained one (4.3) is always extended-real-valued when ˝ ¤ Rn. is reflects the “infinite penalty” for the constraint violation. e next theorem presents basic necessary and sufficient optimality conditions for optimal solutions to the constrained problem (4.3) with a general (convex) cost function. eorem 4.14 Consider the convex problem (4.3) with f Rn R such that f .x/ < for W ! N 1 some point x ˝ under the qualification condition N 2 @1f .x/ N.x ˝/ 0 ; (4.6) N \ NI D f g which automatically holds if f is continuous at x. en the following are equivalent: N (i) x is an optimal solution to (4.3). N (ii) 0 @f.x/ N.x ˝/, i.e., @f.x/ Œ N.x ˝/ . 2 N C NI N \ NI ¤ ; (iii) f .x d/ 0 for all d T.x ˝/. 0 NI 2 NI Proof. Observe first that @ f .x/ 0 and thus (4.6) holds if f is continuous at x. is follows 1 N D f g N from eorem 2.29. To verify now (i) (ii), we use the unconstrained form (4.5) of problem H) (4.3) due to Proposition 4.13. en Proposition 2.35 provides the optimal solution characteriza- tion 0 @g.x/. Applying the subdifferential sum rule from eorem 2.44 under the qualification 2 N condition (4.6) gives us the optimality condition in (ii). To show that (ii) (iii), we have by the tangent-normal duality (3.4) the equivalence H) d T.x ˝/ v; d 0 for all v N.x ˝/: 2 NI ” h i Ä 2 NI Fix an element w @f.x/ with w N.x ˝/. en w; d 0 for all d T.x ˝/ and we 2 N 2 NI h i 2 NI arrive at f .x d/ w; d 0 by eorem 2.84, achieving (iii) in this way. 0 NI h i It remains to justify implication (iii) (i). It follows from Lemma 2.83 that H) f .x d/ f .x/ f 0.x d/ 0 for all d T.x ˝/: N C N NI 2 NI 4.2. OPTIMALITY CONDITIONS 137 By definition of the tangent cone, we have T.x ˝/ clcone.˝ x/, which tells us that d NI D N WD w x T.x ˝/ if w ˝. is yields N 2 NI 2 f .w/ f .x/ f .w x x/ f .x/ f .x d/ f .x/ 0; N D N CN N D N C N which verifies (i) and completes the proof of the theorem. Corollary 4.15 Assume that in the framework of eorem 4.14 the cost function f Rn R is W ! differentiable at x dom f . en the following are equivalent: N 2 (i) x is an optimal solution to problem (4.3). N (ii) f .x/ N.x ˝/, which reduces to f .x/ 0 if x int ˝. r N 2 NI r N D N 2 Proof. Since the differentiability of the convex function f at x implies its continuity and so Lip- N schitz continuity around this point by eorem 2.29. Since @f.x/ f .x/ in this case by N D fr N g eorem 3.3, we deduce this corollary directly from eorem 4.14. Next we consider the following convex optimization problem containing functional in- equality constraints given by finite convex functions, along with the set/geometric constraint as in (4.3). For simplicity, suppose that f is also finite on Rn. Problem .P / is: minimize f .x/ subject to gi .x/ 0 for i 1; : : : ; m; Ä D x ˝: 2 Define the Lagrange function for .P / involving only the cost and functional constraints by m L.x; / f .x/ X i gi .x/ WD C i 1 D with i R and the active index set given by I.x/ i 1; : : : ; m gi .x/ 0 . 2 WD f 2 f g j D g eorem 4.16 Let x be an optimal solution to .P /. en there are multipliers 0 0 and Nm .1; : : : ; m/ R , not equal to zero simultaneously, such that i 0 as i 0; : : : ; m, D 2 D m 0 [email protected]/ X i @gi .x/ N.x ˝/; 2 N C N C NI i 1 D and i gi .x/ 0 for all i 1; : : : ; m. N D D 138 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Proof. Consider the finite convex function '.x/ max ˚f .x/ f .x/; gi .x/ ˇ i 1; : : : ; m« WD N ˇ D and observe that x solves the following problem with no functional constraints: N minimize '.x/ subject to x ˝: 2 Since ' is finite and hence locally Lipschitz continuous, we get from eorem 4.14 that 0 2 @'.x/ N.x ˝/. e subdifferential maximum rule from Proposition 2.54 yields N C NI 0 co @f.x/ Œ [ @gi .x/ N.x ˝/: 2 N [ N C NI i I.x/ 2 N is gives us 0 0 and i 0 for i I.x/ such that 0 Pi I.x/ i 1 and 2 N C 2 N D 0 [email protected]/ X i @gi .x/ N.x ˝/: (4.7) 2 N C N C NI i I.x/ 2 N Letting i 0 for i I.x/, we get i gi .x/ 0 for all i 1; : : : ; m with .0; / 0. WD … N N D D ¤ Many applications including numerical algorithms to solve problem .P / require effective constraint qualification conditions that ensure 0 0 in eorem 4.16. e following one is clas- ¤ sical in convex optimization. Definition 4.17 S’ .CQ/ holds in .P / if there exists u ˝ 2 such that gi .u/ < 0 for all i 1; : : : ; m. D e next theorem provides necessary and sufficient conditions for convex optimization in the qualified, normal, or Karush-Kuhn-Tucker .KKT/ form. eorem 4.18 Suppose that Slater’s CQ holds in .P /. en x is an optimal solution to .P / if N m and only if there exist nonnegative Lagrange multipliers .1; : : : ; m/ R such that 2 m 0 @f.x/ X i @gi .x/ N.x ˝/ (4.8) 2 N C N C NI i 1 D and i gi .x/ 0 for all i 1; : : : ; m. N D D Proof. To verify the necessity part, we only need to show that 0 0 in (4.7). Suppose on the ¤ contrary that 0 0 and find .1; : : : ; m/ 0, vi @gi .x/, and v N.x ˝/ satisfying D ¤ 2 N 2 NI m 0 X i vi v: D C i 1 D 4.2. OPTIMALITY CONDITIONS 139 is readily implies that m m m 0 X i vi ; x x v; x x X i gi .x/ gi .x/ X i gi .x/ for all x ˝; D h Ni C h Ni Ä N D 2 i 1 i 1 i 1 D D D m which contradicts since the Slater condition implies Pi 1 i gi .u/ < 0. D To check the sufficiency of (4.8), take v0 @f.x/, vi @gi .x/, and v N.x ˝/ such that m 2 N 2 N 2 NI 0 v0 P i vi v with i gi .x/ 0 and i 0 for all i 1; : : : ; m. en the optimality D C i 1 C N D D of x in .P / followsD directly from the definitions of the subdifferential and normal cone. Indeed, N for any x ˝ with gi .x/ 0 for i 1; : : : ; m one has 2 Ä D m 0 i vi v; x x X i vi ; x x v; x x D h C Ni D h Ni C h Ni i 1 m D X i Œgi .x/ gi .x/ f .x/ f .x/ Ä N C N i 1 Dm X i gi .x/ f .x/ f .x/ f .x/ f .x/; D C N Ä N i 1 D which completes the proof. Finally in this section, we consider the convex optimization problem .Q/ with inequality and linear equality constraints minimize f .x/ subject to gi .x/ 0 for i 1; : : : ; m; Ä D Ax b; D where A is an p n matrix and b Rp. 2 Corollary 4.19 Suppose that Slater’s CQ holds for problem .Q/ with the set ˝ x Rn Ax WD f 2 j b 0 in Definition 4.17. en x is an optimal solution to .Q/ if and only if there exist nonnegative D g N multipliers 1; : : : ; m such that m 0 @f.x/ X i @gi .x/ im A 2 N C N C i 1 D and i gi .x/ 0 for all i 1; : : : ; m. N D D Proof. e result follows from eorem 4.16 and Proposition 2.12, where the normal cone to the set ˝ is calculated via the image of the adjoint operator A. 140 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS e concluding result presents the classical version of the Lagrange multiplier rule for con- vex problems with differentiable data. Corollary 4.20 Let f and gi for i 1; : : : ; m be differentiable at x and let gi .x/ i I.x/ be D N fr N j 2 N g linearly independent. en x is an optimal solution to problem .P / with ˝ Rn if and only if there N D are nonnegative multipliers 1; : : : ; m such that m 0 f .x/ X i gi .x/ Lx.x; / 2 r N C r N D N i 1 D and i gi .x/ 0 for all i 1; : : : ; m. N D D Proof. In this case N.x ˝/ 0 and it follows from (4.7) that 0 0 contradicts the linear NI D f g D independence of the gradient elements gi .x/ i I.x/ . fr N j 2 N g 4.3 SUBGRADIENT METHODS IN CONVEX OPTIMIZATION is section is devoted to algorithmic aspects of convex optimization and presents two versions of the classical subgradient method: one for unconstrained and the other for constrained optimiza- tion problems. First we study the unconstrained problem: minimize f .x/ subject to x Rn; (4.9) 2 n where f R R is a convex function. Let ˛k as k N be a sequence of positive numbers. e W ! f g 2 n subgradient algorithm generated by ˛k is defined as follows. Given a starting point x1 R , f g 2 consider the iterative procedure: xk 1 xk ˛kvk with some vk @f.xk/; k N : (4.10) C WD 2 2 In this section we assume that problem (4.9) admits an optimal solution and that f is convex and Lipschitz continuous on Rn. Our main goal is to find conditions that ensure the convergence of the algorithm. We split the proof of the main result into several propositions, which are of their own interest. e first one gives us an important subgradient property of convex Lipschitzian functions. Proposition 4.21 Let ` 0 be a Lipschitz constant of f on Rn. For any x Rn, we have the 2 subgradient estimate v ` whenever v @f.x/: k k Ä 2 4.3. SUBGRADIENT METHODS IN CONVEX OPTIMIZATION 141 Proof. Fix x Rn and take any subgradient v @f.x/. It follows from the definition that 2 2 v; u x f .u/ f .x/ for all u Rn : h i Ä 2 Combining this with the Lipschitz property of f gives us v; u x ` u x ; u Rn; h i Ä k k 2 which yields in turn that v; y x x ` y x x ; i.e. v; y ` y ; y Rn : h C i Ä k C k h i Ä k k 2 Using y v, we arrive at the conclusion v `. WD k k Ä n Proposition 4.22 Let ` 0 be a Lipschitz constant of f on R . For the sequence xk generated by f g algorithm (4.10), we have 2 2 2 2 n xk 1 x xk x 2˛k f .xk/ f .x/ ˛k` for all x R ; k N : (4.11) k C k Ä k k C 2 2 Proof. Fix x Rn, k N and get from (4.10) that 2 2 2 2 2 xk 1 x xk ˛kvk x xk x ˛kvk k C k D k 2 k D k 2 k 2 xk x 2˛k vk; xk x ˛ vk : D k k h i C k k k Since vk @f.xk/ and vk ` by Proposition 4.21, it follows that 2 k k Ä vk; x xk f .x/ f .xk/: h i Ä is gives us (4.11) and completes the proof of the proposition. e next proposition provides an estimate of closeness of iterations to the optimal value of (4.9) for an arbitrary sequence ˛k in the algorithm (4.10). Denote f g n V min ˚f .x/ ˇ x R « and Vk min ˚f .x1/; : : : ; f .xk/«: WD ˇ 2 WD Proposition 4.23 Let ` 0 be a Lipschitz constant of f on Rn and let S be the set of optimal solutions to (4.9). For all k N, we have 2 k 2 2 2 d.x1 S/ ` P ˛i I C i 1 0 Vk V D : (4.12) Ä Ä k 2 P ˛i i 1 D 142 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Proof. Substituting any x S into estimate (4.11) gives us N 2 2 2 2 2 xk 1 x xk x 2˛k f .xk/ f .x/ ˛k` : k C Nk Ä k Nk N C Since Vk f .xi / for all i 1; : : : ; k and V f .x/, we get Ä D D N k k 2 2 2 2 xk 1 x x1 x 2 X ˛i f .xi / f .x/ ` X ˛i k C Nk Ä k Nk N C i 1 i 1 Dk k D 2 2 2 x1 x 2 X ˛i .Vk V/ ` X ˛ : Ä k Nk C i i 1 i 1 D D 2 It follows from xk 1 x 0 that k C Nk k k 2 2 2 2 X ˛i Vk V x1 x ` X ˛ : Ä k Nk C i i 1 i 1 D D Since Vk V for every k, we deduce the estimate 2 2 k 2 x1 x ` Pi 1 ˛i 0 Vk V k Nk C D Ä Ä k 2 Pi 1 ˛i D and hence arrive at (4.12) since x S was chosen arbitrarily. N 2 e corollary below shows the number of steps in order to achieve an appropriate error estimate for the optimal value in the case of constant step sizes. Corollary 4.24 Suppose that ˛k for all k N in the setting of Proposition 4.23. en there D 2 exists a positive constant C such that 2 2 0 Vk V < ` whenever k > C= : Ä Proof. From (4.12) we readily get 2 2 2 a k` a ` 2 0 Vk V C with a d.x1 S/ : Ä Ä 2k D 2k C 2 WD I Since a=.2k/ `2=2 < `2 as k > a=.`22/, the conclusion holds with C a=`2. C WD As a direct consequence of Proposition 4.23, we get our first convergence result. Proposition 4.25 In the setting of Proposition 4.23 suppose in addition that 1 X ˛k and lim ˛k 0: (4.13) D 1 k D k 1 !1 D en we have the convergence Vk V as k . ! ! 1 4.3. SUBGRADIENT METHODS IN CONVEX OPTIMIZATION 143 Proof. Given any > 0, choose K N such that ˛i ` < for all i K. For any k > K, using 2 (4.12) ensures the estimates K k 2 2 2 2 2 d.x1 S/ ` P ˛i ` P ˛i I C i 1 i K 0 Vk V D D Ä Ä k C k 2 P ˛i 2 P ˛i i 1 i 1 D K kD 2 2 2 d.x1 S/ ` P ˛i P ˛i I C i 1 i K D D Ä k C k 2 P ˛i 2 P ˛i i 1 i 1 D K D 2 2 2 d.x1 S/ ` P ˛i I C i 1 D =2: Ä k C 2 P ˛i i 1 D k e assumption Pk1 1 ˛k yields limk Pi 1 ˛i , and hence D D 1 !1 D D 1 0 lim inf.Vk V/ lim sup.Vk V/ =2: Ä k Ä k Ä !1 !1 Since > 0 is chosen arbitrarily, one has limk .Vk V/ 0 or Vk V as k . !1 D ! ! 1 e next result gives us important information about the value convergence of the subgra- dient algorithm (4.10). Proposition 4.26 In the setting of Proposition 4.25 we have lim inf f .xk/ V: k D !1 Proof. Since V is the optimal value in (4.9), we get f .xk/ V for any k N. is yields 2 lim inf f .xk/ V: k !1 To verify the opposite inequality, suppose on the contrary that lim infk f .xk/ > V and then find a small number > 0 such that !1 lim inf f .xk/ 2 > V min f .x/ f .x/; k D x Rn D N !1 2 where x is an optimal solution to problem (4.9). us N lim inf f .xk/ > f .x/ : k N C !1 144 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS In the case where lim infk f .xk/ < , by the definition of lim inf, there is k0 N such that !1 1 2 for every k k0 we have that f .xk/ > lim inf f .xk/ ; k !1 and hence f .xk/ f .x/ > for every k k0. Note that this conclusion also holds if N lim infk f .xk/ . It follows from (4.11) with x x that !1 D 1 DN 2 2 2 2 xk 1 x xk x 2˛k f .xk/ f .x/ ˛k` for every k k0: (4.14) k C Nk Ä k Nk N C 2 Let k1 be such that ˛k` < for all k k1 and let k1 max k1; k0 . For any k > k1, it easily N WD f g N follows from estimate (4.14) that k 2 2 2 x x x x ˛ ::: x x X ˛j ; k 1 k k k1 k C Nk Ä k Nk Ä Ä N N j k1 D N which contradicts the assumption Pk1 1 ˛k in (4.13) and so completes the proof. D D 1 Now we are ready to justify the convergence of the subgradient algorithm (4.10). eorem 4.27 Suppose that the generating sequence ˛k in (4.10) satisfies f g 1 1 2 X ˛k and X ˛ < : (4.15) D 1 k 1 k 1 k 1 D D en the sequence Vk converges to the optimal value V and the sequence of iterations xk in f g f g (4.10) converges to some optimal solution x of (4.9). N Proof. Since (4.15) implies (4.13), the sequence Vk converges to V by Proposition 4.25. It f g remains to prove that xk converges to some optimal solution x of problem (4.9). Let ` 0 be f g N a Lipschitz constant of f on Rn. Since the set S of optimal solutions to (4.9) is assumed to be nonempty, we pick any x S and substitute x x in estimate (4.11) of Proposition 4.22. It gives 2 D us e e 2 2 2 2 xk 1 x xk x 2˛k f .xk/ V ˛k` for all k N : (4.16) k C ek Ä k ek C 2 Since f .xk/ V 0, inequality (4.16) yields 2 2 2 2 xk 1 x xk x ˛k` : k C ek Ä k ek C en we get by induction that k 2 2 2 2 xk 1 x x1 x ` X ˛i : (4.17) k C k Ä k k C e e i 1 D 4.3. SUBGRADIENT METHODS IN CONVEX OPTIMIZATION 145 2 e right-hand side of (4.17) is finite by our assumption that Pk1 1 ˛k < , and hence D 1 sup xk 1 x < : k N k C ek 1 2 is implies that the sequence xk is bounded. Furthermore, Proposition 4.26 tells us that f g lim inf f .xk/ V: k D !1 Let xk be a subsequence of xk such that f j g f g lim f .xk / V: (4.18) j j D !1 en xk is also bounded and contains a limiting point x. Without loss of generality assume f j g N that xk x as j . We directly deduce from (4.18) and the continuity of f that x is an j !N ! 1 N optimal solution to (4.9). Fix any j N and any k kj . us we can rewrite (4.16) with x x 2 DN and see that e k 2 2 2 2 2 2 2 xk 1 x xk x ˛k` ::: xkj x ` X ˛i : k C Nk Ä k Nk C Ä Ä N C i kj D Taking first the limit as k and then the limit as j in the resulting inequality above ! 1 ! 1 gives us the estimate 2 2 2 1 2 lim sup xk 1 x lim xkj x ` lim X ˛i : k k C Nk Ä j N C j !1 !1 i kj !1 D 2 Since xkj x and Pk1 1 ˛k < , this implies that !N D 1 2 lim sup xk 1 x 0; k k C Nk D !1 and thus justifies the claimed convergence of xk x as k . !N ! 1 Finally in this section, we present a version of the subgradient algorithm for convex prob- lems of constrained optimization given by minimize f .x/ subject to x ˝; (4.19) 2 where f Rn R is a convex function that is Lipschitz continuous on Rn with some constant W ! ` 0, while—in contrast to (4.9)—we have now the constraint on x given by a nonempty, closed, convex set ˝ Rn. e following counterpart of the subgradient algorithm (4.10) for constrained problems is known as the projected subgradient method. 146 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Given a sequence of positive numbers ˛k and given a starting point x1 ˝, the sequence f g 2 of iterations xk is constructed by f g xk 1 ˘.xk ˛kvk ˝/ with some vk @f.xk/; k N; (4.20) C WD I 2 2 where ˘.x ˝/ stands for the (unique) Euclidean projection of x onto ˝. I e next proposition, which is a counterpart of Proposition 4.22 for algorithm (4.20), plays a major role in justifying convergence results for the projected subgradient method. Proposition 4.28 In the general setting of (4.20) we have the estimate 2 2 2 2 xk 1 x xk x 2˛k f .xk/ f .x/ ˛k` for all x ˝; k N : k C k Ä k k C 2 2 Proof. Define zk 1 xk ˛kvk and fix any x ˝. It follows from Proposition 4.22 that C WD 2 2 2 2 2 zk 1 x xk x 2˛k f .xk/ f .x/ ˛k` for all k N : (4.21) k C k Ä k k C 2 Now by projecting zk 1 onto ˝ and using Proposition 1.79, we get C xk 1 x ˘.zk 1 ˝/ ˘.x ˝/ zk 1 x : k C k D k C I I k Ä k C k Combining this with estimate (4.21) gives us 2 2 2 2 xk 1 x xk x 2˛k f .xk/ f .x/ ˛k` for all k N; k C k Ä k k C 2 which thus completes the proof of the proposition. Based on this proposition and proceeding in the same way as for the subgradient algorithm (4.10) above, we can derive similar convergence results for the projected subgradient algorithm (4.20) under the assumptions (4.13) on the generating sequence ˛k . f g 4.4 THE FERMAT-TORRICELLI PROBLEM In 1643 Pierre de Fermat proposed to Evangelista Torricelli the following optimization problem: Given three points in the plane, find another point such that the sum of its distances to the given points is minimal. is problem was solved by Torricelli and was named the Fermat-Torricelli problem. A more general version of the Fermat-Torricelli problem asks for a point that minimizes the sum of the distances to a finite number of given points in Rn. is is one of the main problems in location science, which has been studied by many researchers. e first numerical algorithm for solving the general Fermat-Torricelli problem was introduced by Endre Weiszfeld in 1937. As- sumptions that guarantee the convergence of the Weiszfeld algorithm were given by Harold Kuhn in 1972. Kuhn also pointed out an example in which the Weiszfeld algorithm fails to converge. Several new algorithms have been introduced recently to improve the Weiszfeld algorithm. e 4.4. THE FERMAT-TORRICELLI PROBLEM 147 goal of this section is to revisit the Fermat-Torricelli problem from both theoretical and numerical viewpoints by using techniques of convex analysis and optimization. n Given points ai R as i 1; : : : ; m, define the (nonsmooth) convex function 2 D m '.x/ X x ai : (4.22) WD k k i 1 D en the mathematical description of the Fermat-Torricelli problem is minimize '.x/ over all x Rn: (4.23) 2 Proposition 4.29 e Fermat-Torricelli problem (4.23) has at least one solution. Proof. e conclusion follows directly from eorem 4.10. Proposition 4.30 Suppose that ai as i 1; : : : ; m are not collinear, i.e., they do not belong to the D same line. en the function ' in (4.22) is strictly convex, and hence the Fermat-Torricelli problem (4.23) has a unique solution. Proof. Since each function 'i .x/ x ai as i 1; : : : ; m is obviously convex, their sum m WD k k nD ' Pi 1 'i is convex as well, i.e., for any x; y R and .0; 1/ we have D D 2 2 ' x .1 /y '.x/ .1 /'.y/: (4.24) C Ä C Supposing by contradiction that ' is not strictly convex, find x; y Rn with x y and .0; 1/ N N 2 N ¤ N 2 such that (4.24) holds as equality. It follows that 'i x .1 /y 'i .x/ .1 /'i .y/ for all i 1; : : : ; m; N C N D N C N D which ensures therefore that .x ai / .1 /.y ai / .x ai / .1 /.y ai / ; i 1; : : : ; m: k N C N k D k N k C k N k D If x ai and y ai , then there exists ti > 0 such that ti .x ai / .1 /.y ai /, and hence N ¤ N ¤ N D N 1 x ai i .y ai / with i : N D N WD ti Since x y, we have i 1. us N ¤ N ¤ 1 i ai x y L.x; y/; D 1 i N 1 i N 2 N N where L.x; y/ signifies the line connecting x and y. Both cases where x ai and y ai give us N N N N N D N D ai L.x; y/. Hence ai L.x; y/ for i 1; : : : ; m, a contradiction. 2 N N 2 N N D 148 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS We proceed further with solving the Fermat-Torricelli problem for three points in Rn. Let us first present two elementary lemmas. n Lemma 4.31 Let a1 and a2 be any two unit vectors in R . en we have a1 a2 IB if and 1 2 only if a1; a2 . h i Ä 2 Proof. If a1 a2 IB, then a1 a2 IB and 2 C 2 a1 a2 1: k C k Ä By squaring both sides, we get the inequality 2 a1 a2 1; k C k Ä which subsequently implies that 2 2 a1 2 a1; a2 a2 2 2 a1; a2 1: k k C h i C k k D C h i Ä 1 erefore, a1; a2 . e proof of the converse implication is similar. h i Ä 2 n Lemma 4.32 Let a1; a2; a3 R with a1 a2 a3 1. en a1 a2 a3 0 if and 2 k k D k k D k k D C C D only if we have the equalities 1 a1; a2 a2; a3 a3; a1 : (4.25) h i D h i D h i D 2 Proof. It follows from the condition a1 a2 a3 0 that C C D 2 a1 a1; a2 a1; a3 0; (4.26) k k C h i C h i D 2 a2; a1 a2 a2; a3 0; (4.27) h i C k k C h i D 2 a3; a1 a3; a2 a3 0: (4.28) h i C h i C k k D Subtracting (4.28) from (4.26) gives us a1; a2 a2; a3 and similarly a2; a3 a3; a1 . Sub- h i D h i h i D h i stituting these equalities into (4.26), (4.27), and (4.28) ensures the validity of (4.25). Conversely, assume that (4.25) holds. en we have 3 3 2 2 a1 a2 a3 X ai X ai ; aj 0; k C C k D k k C h i D i 1 i;j 1;i j D D ¤ which readily yields the equality a1 a2 a3 0 and thus completes the proof. C C D 4.4. THE FERMAT-TORRICELLI PROBLEM 149 Now we use subdifferentiation of convex functions to solve the Fermat-Torricelli problem for three points in Rn. Given two nonzero vectors u; v Rn, define 2 u; v cos.u; v/ h i WD u v k k k k and also consider, when x ai , the three vectors N ¤ x ai vi N ; i 1; 2; 3: D x ai D k N k Geometrically, each vector vi is a unit one pointing in the direction from the vertex ai to x. N Observe that the classical (three-point) Fermat-Torricelli problem always has a unique solution even if the given points belong to the same line. Indeed, in the latter case the middle point solves the problem. e next theorem fully describes optimal solutions of the Fermat-Torricelli problem for three points in two distinct cases. eorem 4.33 Consider the Fermat-Torricelli problem (4.23) for the three points a1; a2; a3 2 Rn. e following assertions hold: (i) Let x a1; a2; a3 . en x is the solution of the problem if and only if N … f g N cos.v1; v2/ cos.v2; v3/ cos.v3; v1/ 1=2: D D D (ii) Let x a1; a2; a3 , say x a1. en x is the solution of the problem if and only if N 2 f g N D N cos.v2; v3/ 1=2: Ä Proof. In case (i) function (4.22) is differentiable at x. en x solves (4.23) if and only if N N '.x/ v1 v2 v3 0: r N D C C D By Lemma 4.32 this holds if and only if vi ; vj cos.vi ; vj / 1=2 for any i; j ˚1; 2; 3« with i j: h i D D 2 ¤ To verify (ii), employ the subdifferential Fermat rule (2.35) and sum rule from Corollary 2.45, which tell us that x a1 solves the Fermat-Torricelli problem if and only if N D 0 @'.a1/ v2 v3 IB: 2 D C C Since v2 and v3 are unit vectors, it follows from Lemma 4.31 that v2; v3 cos.v2; v3/ 1=2; h i D Ä which completes the proof of the theorem. 150 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Figure 4.2: Construction of the Torricelli point. Example 4.34 is example illustrates geometrically the solution construction for the Fermat- Torricelli problem in the plane. Consider (4.23) with the three given points A, B, and C on R2 as shown in Figure 4.2. If one of the angles of the triangle ABC is greater than or equal to 120ı, then the corresponding vertex is the solution to (4.23) by eorem 4.33(ii). Consider the case where none of the angles of the triangle is greater than or equal to 120ı. Construct the two equilateral triangles ABD and ACE and let S be the intersection of DC and BE as in the figure. Two quadrilaterals ADBC and ABCE are convex, and hence S lies inside the triangle ABC . It is clear that two triangles DAC and BAE are congruent. A rotation of 60ı about A maps the triangle DAC to the triangle BAE. e rotation maps CD to BE, so DSB 60 . Let T be the image of † D ı S though this rotation. en T belongs to BE. It follows that AST ASE 60 . Moreover, † D† D ı DSA 60 , and hence BSA 120 . It is now clear that ASC 120 and BSC 120 . † D ı † D ı † D ı † D ı By eorem 4.33(i) the point S is the solution of the problem under consideration. Next we present the Weiszfeld algorithm [32] to solve numerically the Fermat-Torricelli problem in the general case of (4.23) for m points in Rn that are not collinear. is is our standing assumption in the rest of this section. To proceed, observe that the gradient of the function ' from (4.22) is calculated by m x ai '.x/ X for x ˚a1; : : : ; am«: r D x ai … i 1 D k k Solving the gradient equation '.x/ 0 gives us r D m ai Pi 1 D x ai x k k f .x/: (4.29) D m 1 DW Pi 1 D x ai k k 4.4. THE FERMAT-TORRICELLI PROBLEM 151 n We define f .x/ on the whole space R by letting f .x/ x for x a1; : : : ; am . WD 2 f g n Weiszfeld algorithm consists of choosing an arbitrary starting point x1 R , taking f is 2 from (4.29), and defining recurrently xk 1 f .xk/ for k N; (4.30) C WD 2 where f .x/ is called the algorithm mapping. In the rest of this section we present the precise conditions and the proof of convergence for Weiszfeld algorithm in the way suggested by Kuhn [10] who corrected some errors from [32]. In contrast to [10], we employ below subdifferential rules of convex analysis, which allow us to simplify the proof of convergence. e next proposition ensures decreasing the function value of ' after each iteration, i.e., the descent property of the algorithm. Proposition 4.35 If f .x/ x, then '.f .x// < '.x/. ¤ Proof. e assumption of f .x/ x tells us that x is not a vertex from a1; : : : ; am . Observe ¤ f g also that the point f .x/ is a unique minimizer of the strictly convex function m 2 z ai g.z/ X k k WD x ai i 1 D k k and that g.f .x// < g.x/ '.x/ since f .x/ x. We have furthermore that D ¤ m 2 f .x/ ai g f .x/ X k k D x ai i 1 k k Dm 2 . x ai f .x/ ai x ai / X k k C k k k k D x ai i 1 k k D m 2 . f .x/ ai x ai / '.x/ 2 '.f .x// '.x/ X k k k k : D C C x ai i 1 D k k is clearly implies the inequality m 2 . f .x/ ai x ai / 2' f .x/ X k k k k < 2'.x/ C x ai i 1 D k k and thus completes the proof of the proposition. e following two propositions show how the algorithm mapping f behaves near a vertex that solves the Fermat-Torricelli problem. First we present a necessary and sufficient condition ensuring the optimality of a vertex in (4.23). Define m ai aj Rj X ; j 1; : : : ; m: WD ai aj D i 1;i j D ¤ k k 152 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Proposition 4.36 e vertex aj solves problem (4.23) if and only if Rj 1. k k Ä Proof. Employing the Fermat rule from Proposition 2.35 and the subdifferential sum rule from Corollary 2.45 shows that the vertex aj solves problem (4.23) if and only if 0 @'.aj / Rj IB.0; 1/; 2 D C which is clearly equivalent to the condition Rj 1. k k Ä Proposition 4.36 from convex analysis allows us to essentially simplify the proof of the following result established by Kuhn. We use the notation: s s 1 0 n f .x/ f f .x/ for s N with f .x/ x; x R : WD 2 WD 2 Proposition 4.37 Suppose that the vertex aj does not solve (4.23). en there is ı > 0 such that the condition 0 < x aj ı implies the existence of s N with k k Ä 2 s s 1 f .x/ aj > ı and f .x/ aj ı: (4.31) k k k k Ä Proof. For any x a1; : : : ; am , we have from (4.29) that … f g m ai aj Pi 1;i j D ¤ x ai f .x/ aj k k and so D m 1 Pi 1 D x ai k k m ai aj Pi 1;i j f .x/ aj D ¤ x ai lim lim k k Rj ; j 1; : : : ; m: x aj x a D x aj x aj D D ! j ! m k k 1 Pi 1;i j k k C D ¤ x ai k k It follows from Proposition 4.36 that f .x/ aj lim k k Rj > 1; j 1; : : : ; m; (4.32) x aj x aj D k k D ! k k which allows us to select > 0; ı > 0 such that f .x/ aj k k > 1 whenever 0 < x aj ı: x aj C k k Ä k k 4.4. THE FERMAT-TORRICELLI PROBLEM 153 q 1 us for every s N the validity of 0 < f .x/ aj ı with any q 1; : : : ; s yields 2 k k Ä D s s 1 s f .x/ aj .1 / f .x/ aj ::: .1 / x aj : k k C k k C k k s Since .1 / x aj as s , it gives us (4.31) and completes the proof. C k k ! 1 ! 1 We finally present the main result on the convergence of the Weiszfeld algorithm. eorem 4.38 Let xk be the sequence generated by Weiszfeld algorithm (4.30) and let xk f g … a1; : : : ; am for k N. en xk converges to the unique solution x of (4.23). f g 2 f g N Proof. If xk xk 1 for some k k0, then xk is a constant sequence for k k0, and hence D C D f g it converges to xk . Since f .xk / xk and xk is not a vertex, xk solves the problem. us 0 0 D 0 0 0 we can assume that xk 1 xk for every k. It follows from Proposition 4.35 that the sequence C ¤ '.xk/ of nonnegative numbers is decreasing, so it converges. is gives us f g lim '.xk/ '.xk 1/ 0: (4.33) k C D !1 By the construction of xk , we have that each xk belongs to the compact set co a1; : : : ; am . f g f g is allows us to select from xk a subsequence xk converging to some point z as . f g f g N ! 1 We have to show that z x, i.e., it is the unique solution to (4.23). Indeed, we get from (4.33) N DN that lim '.xk / ' f .xk / 0; D !1 which implies by the continuity of the functions ' and f that '.z/ '.f .z//, i.e., f .z/ z and N D N N DN thus z solves (4.23) if it is not a vertex. N It remains to consider the case where z is a vertex, say a1. Suppose by contradiction that z N N D a1 x and select ı > 0 sufficiently small so that (4.31) holds for some s N and that IB.a1 ı/ ¤ N 2 I does not contain x and ai for i 2; : : : ; m. Since xk z a1, we can assume without loss of N D !N D generality that this sequence is entirely contained in IB.a1 ı/. For x xk select l1 so that xl I D 1 1 2 IB.a1 ı/ and f .xl / IB.a1 ı/. Choosing now k > l1 and applying Proposition 4.37 give us I 1 … I l2 > l1 with xl IB.a1 ı/ and f .xl / IB.a1 ı/. Repeating this procedure, find a subsequence 2 2 I 2 … I xl such that xl IB.a1 ı/ while f .xl / is not in this ball. Extracting another subsequence f g 2 I if needed, we can assume that xl converges to some point q satisfying f .q/ q. If q is not a f g N N DN N vertex, then it solves (4.23) by the above, which is a contradiction because the solution x is not N in IB.a1 ı/. us q is a vertex, which must be a1 since the other vertexes are also not in the ball. I N en we have f .xl / a1 lim k k ; xl a1 D 1 !1 k k which contradicts (4.32) in Proposition 4.37 and completes the proof of the theorem. 154 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS 4.5 A GENERALIZED FERMAT-TORRICELLI PROBLEM ere are several generalized versions of the Fermat-Torricelli problem known in the literature under different names; see, e.g., our papers [15, 16, 17] and the references therein. Note that the constrained version of the problem below is also called a “generalized Heron problem” after Heron from Alexandria who formulated its simplest version in the plane in the 1st century AD. Following [15, 16], we consider here a generalized Fermat-Torricelli problem defined via the minimal time functions (3.9) for finitely many closed, convex target sets in Rn with a constraint set of the same structure. In [6, 11, 29] and their references, the reader can find some particular versions of this problem and related material on location problems. We also refer the reader to [15, 17] to nonconvex problems of this type. Let the target sets ˝i for i 1; : : : ; m and the constraint set ˝0 be nonempty, closed, con- n D F vex subsets of R . Consider the minimal time functions TF .x ˝i / T .x/ from (3.9) for ˝i , I D ˝i i 1; : : : ; m, with the constant dynamics defined by a closed, bounded, convex set F Rn that D contains the origin as an interior point. en the generalized Fermat-Torricelli problem is formu- lated as follows: m minimize H.x/ X TF .x ˝i / subject to x ˝0: (4.34) WD I 2 i 1 D Our first goal in this section is to establish the existence and uniqueness of solutions to the opti- mization problem (4.34). We split the proof into several propositions of their own interest. Let us start with the following property of the Minkowski gauge from (3.11). Proposition 4.39 Assume that F is a closed, bounded, convex subset of Rn such that 0 int F . en 2 F .x/ 1 if and only if x bd F . D 2 Proof. Suppose that F .x/ 1. en there exists a sequence of real numbers tk that converges D f g to t 1 and such that x tkF for every k. Since F is closed, it yields x F . To complete the D 2 2 proof of the implication, we show that x int F . By contradiction, assume that x int F and … 1 2 1 choose t > 0 so small that x tx F . en x .1 t/ F . is tells us F .x/ .1 t/ < C 2 2 C Ä C 1, a contradiction. Now let x bd F . Since F is closed, we have x F , and hence F .x/ 1. Suppose on the 2 2 Ä contrary that F .x/ < 1. en there exists a sequence of real numbers tk that converges WD f g to with x tkF for every k. en x F F .1 /0 F . By Proposition 3.32, the 2 2 D C n Minkowski gauge F is continuous, so the set u R F .u/ < 1 is an open subset of F f 2 j g containing x. us x int F , which completes the proof. 2 Recall further that a set Q Rn is said to be strictly convex if for any x; y Q with x y 2 ¤ and for any t .0; 1/ we have tx .1 t/y int Q. 2 C 2 4.5. A GENERALIZED FERMAT-TORRICELLI PROBLEM 155 Proposition 4.40 Assume that F is a closed, bounded, strictly convex subset of Rn and such that 0 int F . en 2 F .x y/ F .x/ F .y/ with x; y 0 (4.35) C D C ¤ if and only if x y for some > 0. D Proof. Denote ˛ F .x/, ˇ F .y/ and observe that ˛; ˇ > 0 since F is bounded. If the WD WD linearity property (4.35) holds, then x y F C Á 1; ˛ ˇ D C which implies in turn that x ˛ y ˇ F Á 1: ˛ ˛ ˇ C ˇ ˛ ˇ D C C is allows us to conclude that x ˛ y ˇ bd F: ˛ ˛ ˇ C ˇ ˛ ˇ 2 C C x y ˛ Since F , F , and .0; 1/, it follows from the strict convexity of F that ˛ 2 ˇ 2 ˛ ˇ 2 C x y F .x/ ; i.e., x y with : ˛ D ˇ D D F .y/ e opposite implication in the proposition is obvious. Proposition 4.41 Let F be as in Proposition 4.40 and let ˝ be a nonempty, closed, convex subset of n n R . For any x R , the set ˘F .x ˝/ is a singleton. 2 I Proof. We only need to consider the case where x ˝. It is clear that ˘F .x ˝/ . Suppose … I ¤ ; by contradiction that there are u1; u2 ˘F .x ˝/ with u1 u2. en 2 I ¤ TF .x ˝/ F .u1 x/ F .u2 x/ r > 0; I D D D u1 x u2 x which implies that bd F and bd F . Since F is strictly convex, we have r 2 r 2 1 u1 x 1 u2 x 1 u1 u2 C xÁ int F; 2 r C 2 r D r 2 2 u1 u2 and so we get F .u x/ < r TF .x ˝/ with u C ˝. is contradiction completes D I WD 2 2 the proof of the proposition. 156 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS In what follows we identify the generalized projection ˘F .x ˝/ with its unique element I when both sets F and ˝ are strictly convex. Proposition 4.42 Let F be a closed, bounded, convex subset of Rn such that 0 int F and let ˝ be n 2 a nonempty, closed set ˝ R with x ˝. If ! ˘F .x ˝/, then ! bd ˝. N … N 2 NI N 2 Proof. Suppose that ! int ˝. en there is > 0 such that IB.! / ˝. Define N 2 N I x ! z ! N N IB.! /: WD N C x ! 2 N I k N N k and observe from the construction of the Minkowski gauge (3.11) that x ! F .z x/ F ! N N xÁ 1 ÁF .! x/ < F .! x/ TF .x ˝/: N D N C x ! N D x ! N N N N D NI k N N k k N N k is shows that ! must be a boundary point of ˝. N Figure 4.3: ree sets that satisfy assumptions of Proposition 4.43. Proposition 4.43 In the setting of problem (4.34) suppose that the sets ˝i for i 1; : : : ; m are D strictly convex. If for any x; y ˝0 with x y there is i0 1; : : : ; m such that 2 ¤ 2 f g L.x; y/ ˝i \ 0 D; for the line L.x; y/ connecting x and y, then the function m H.x/ X TF .x ˝i / D I i 1 D defined in (4.34) is strictly convex on the constraint set ˝0. 4.5. A GENERALIZED FERMAT-TORRICELLI PROBLEM 157 Proof. Suppose by contradiction that there are x; y ˝0, x y, and t .0; 1/ with 2 ¤ 2 H tx .1 t/y tH.x/ .1 t/H.y/: C D C Using the convexity of each function TF .x ˝i / for i 1; : : : ; m, we have I D TF tx .1 t/y ˝i tTF .x ˝i / .1 t/TF .y ˝i /; i 1; : : : ; m: (4.36) C I D I C I D Suppose further that L.x; y/ ˝i for some i0 1; : : : ; m and define \ 0 D; 2 f g u ˘F .x ˝i / and v ˘F .y ˝i /: WD I 0 WD I 0 en it follows from (4.36), eorem 3.33, and the convexity of the Minkowski gauge that tF .u x/ .1 t/F .v y/ tTF .x ˝i / .1 t/TF .y ˝i / C D I 0 C I 0 TF tx .1 t/y ˝i D C I 0 F tu .1 t/v .tx .1 t/y/ Ä C C tF .u x/ .1 t/F .v y/; Ä C which shows that tu .1 t/v ˘F .tx .1 t/y ˝i /. us u v, since otherwise we have C D C I 0 D tu .1 t/v int ˝i , a contradiction. is gives us u v ˘F .tx .1 t/y ˝i /. Apply- C 2 0 D D C I 0 ing (4.36) again tells us that F u .tx .1 t/y F t.u x/ F .1 t/.u y/; C D C so u x; u y 0 due to x; y ˝i . By Proposition 4.40, we find > 0 such that t.u x/ ¤ … 0 D .1 t/.u y/, which yields .1 t/ u x .u y/ with 1: D D t ¤ is justifies the inclusion 1 u x y L.x; y/; D 1 1 2 which is a contradiction that completes the proof of the proposition. Now we are ready to establish the existence and uniqueness theorem for (4.34). eorem 4.44 In the setting of Proposition 4.43 suppose further that at least one of the sets among ˝i for i 0; : : : ; m is bounded. en the generalized Fermat-Torricelli problem (4.34) D has a unique solution. 158 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Proof. By the strict convexity of H from Proposition 4.43, it suffices to show that problem (4.34) admits an optimal solution. If ˝0 is bounded, then it is compact and the conclusion follows from the (Weierstrass) eorem 4.10 since the function H in (4.34) Lipschitz continuous by Corollary 3.34. Consider now the case where one of the sets ˝i for i 1; : : : ; m is bounded; say D it is ˝1. In this case we get that for any ˛ R the set 2 L˛ ˚x ˝0 ˇ H.x/ ˛« ˚x ˝0 ˇ TF .x ˝1/ ˛« ˝0 .˝1 ˛F / D 2 ˇ Ä 2 ˇ I Ä \ is compact. is ensures the existence of a minimizer for (4.34) by eorem 4.10. Next we completely solve by means of convex analysis and optimization the following sim- plified version of problem (4.34): 3 2 minimize H.x/ X d.x ˝i /; x R ; (4.37) D I 2 i 1 D 2 where the target sets ˝i IB.ci ri / for i 1; 2; 3 are the closed balls of R whose centers ci are WD I D supposed to be distinct from each other to avoid triviality. We first study properties of this problem that enable us to derive verifiable conditions in terms of the initial data .ci ; ri / ensuring the uniqueness of solutions to (4.37). Proposition 4.45 e solution set S of (4.37) is a nonempty, compact, convex set in R2. Proof. e existence of solutions (4.37) follows from the proof of eorem 4.44. It is also easy to see that S is closed and bounded. e convexity of S can be deduced from the convexity of the distance functions d.x ˝i /. I To proceed, for any u R2 define the index subset 2 I.u/ ˚i 1; 2; 3 ˇ u ˝i «; (4.38) WD 2 f g ˇ 2 which is widely used in what follows. Proposition 4.46 Suppose that x does not belong to any of the sets ˝i , i 1; 2; 3, i.e., I.x/ . N D N D; en x is an optimal solution to (4.37) if and only if x solves the classical Fermat-Torricelli problem N N (4.23) generated by the centers of the balls ai ci for i 1; 2; 3 in (4.22). WD D Proof. Assume that x is a solution to (4.37) with x ˝i for i 1; 2; 3. Choose ı > 0 such that N N … D IB.x ı/ ˝i as i 1; 2; 3 and get following for every x IB.x ı/: NI \ D; D 2 NI 3 3 3 3 X x ci X ri inf H.x/ H.x/ H.x/ X x ci X ri : k N k D 2 D N Ä D k k i 1 i 1 x R i 1 i 1 D D 2 D D 4.5. A GENERALIZED FERMAT-TORRICELLI PROBLEM 159 is shows that x is a local minimum of the problem N 3 2 minimize F.x/ X x ci ; x R ; (4.39) WD k k 2 i 1 D so it is a global absolute minimum of this problem due to the convexity of F. us x solves the N classical Fermat-Torricelli problem (4.23) generated by c1; c2; c3. e converse implication is also straightforward. e next result gives a condition for the uniqueness of solutions to (4.37) in terms of the index subset defined in (4.38). Proposition 4.47 Let x be a solution to (4.37) such that I.x/ in (4.38). en x is the unique N N D; N solution to (4.37), i.e., S x . D f Ng Proof. Suppose by contradiction that there is u S such that u x. Since I.x/ , we have 2 ¤ N N D; x ˝i for i 1; 2; 3. en Proposition 4.46 tells us that x solves the classical Fermat-Torricelli N … D N problem generated by c1; c2; c3. It follows from the convexity of S in Proposition 4.45 that Œx; u S. us we find v x such that v S and v ˝i for i 1; 2; 3. It shows that v also N ¤ N 2 … D solves the classical Fermat-Torricelli problem (4.39). is contradicts the uniqueness result for (4.39) and hence justifies the claim of the proposition. e following proposition verifies a visible geometric property of ellipsoids. Proposition 4.48 Given ˛ > 0 and a; b R2 with a b, consider the set 2 ¤ E ˚x R2ˇ x a x b ˛« WD 2 ˇ k k C k k D and suppose that a b < ˛. For x; y E with x y, we have k k 2 ¤ x y x y C a C b < ˛; 2 C 2 x y which ensures, in particular, that C E. 2 … Proof. It follows from the triangle inequality that x y x y 1 C a C b x a y a x b y b Á 2 C 2 D2 k C k C k C k 1 x a y a x b y b Á ˛: Ä2 k k C k k C k k C k k D 160 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Suppose by contradiction that x y x y C a C b ˛; (4.40) 2 C 2 D x y which means that C E. Since our norm is (always) Euclidean, it gives 2 2 x a ˇ.y a/ and x b .y b/ D D for some numbers ˇ; .0; / 1 since x y. is implies in turn that 2 1 n f g ¤ 1 ˇ 1 a x y L.x; y/ and b x y L.x; y/: D 1 ˇ 1 ˇ 2 D 1 1 2 x y Since ˛ > a b , we see that x; y L.a; b/ Œa; b. To check now that C Œa; b, impose k k 2 n 2 2 x y without loss of generality the following order of points: x, C , a; b, and y. en 2 x y x y C a C b < x a x b ˛; 2 C 2 k k C k k D x y which contradicts (4.40). Since C Œa; b, it yields 2 2 x y x y C a C b a b < ˛; 2 C 2 D k k x y which is a contradiction. us C E and the proof is complete. 2 … e next result ensures the uniqueness of solutions to (4.37) via the index set (4.38) differ- ently from Proposition 4.47. Proposition 4.49 Let S be the solution set for problem (4.37) and let x S satisfy the conditions: N 2 I.x/ i and x Œcj ; ck, where i; j; k are distinct indices in 1; 2; 3 . en S is a singleton, namely N D f g N … f g S x . D f Ng Proof. Suppose for definiteness that I.x/ 1 . en we have x ˝1, x ˝2, x ˝3, and N D f g N 2 N … N … x Œc2; c3. Denote N … ˛ x c2 x c3 inf H.x/ r2 r3 WD k N k C k N k D x R2 C C 2 and observe that ˛ > c2 c3 . Consider the ellipse k k 2 E ˚x R ˇ x c2 x c3 ˛« WD 2 ˇ k k C k k D 4.5. A GENERALIZED FERMAT-TORRICELLI PROBLEM 161 for which we get x E ˝1. Arguing by contradiction, suppose that S is not a singleton, i.e., N 2 \ there is u S with u x. e convexity of S implies that Œx; u S. We can choose v Œx; u 2 ¤ N N 2 N sufficiently close to but different from x so that Œx; v ˝2 and Œx; v ˝3 . Let us first N N \ D; N \ D; show that v ˝1. Assuming the contrary gives us d.v ˝1/ 0, and hence … I D H.v/ inf H.x/ d.v ˝2/ d.v ˝3/ v c2 r2 v c3 r3; D x R2 D I C I D k k C k k 2 x v which implies that v E. It follows from the convexity of S and ˝1 that N C S ˝1. We 2 2 2 \ x v x v x v have furthermore that N C ˝2 and N C ˝3, which tell us that N C E. But this 2 … 2 … 2 2 cannot happen due to Proposition 4.48. erefore, v S and I.v/ , which contradict the 2 D; result from Proposition 4.47 and thus justify that S is a singleton. Figure 4.4: A Fermat-Torricelli problem with I.x/ 2. j N j D Now we verify the expected boundary property of optimal solutions to (4.37). Proposition 4.50 Let x S and let I.x/ i; j 1; 2; 3 . en x bd .˝i ˝j /. N 2 N D f g f g N 2 \ Proof. Assume for definiteness I.x/ 1; 2 , i.e., x ˝1 ˝2 and x ˝3. Arguing by con- N D f g N 2 \ N … tradiction, suppose that x int .˝1 ˝2/, and therefore there exists ı > 0 such that IB.x ı/ N 2 \ NI ˝1 ˝2. Denote p ˘.x ˝3/, x p > 0 and take q Œx; p IB.x; ı/ satisfying the \ WD NI WD k N k 2 N \ N condition q p < x p d.x ˝3/. en we get k k k N k D NI 3 3 H.q/ X d.q ˝i / d.q ˝3/ q p < x p d.x ˝3/ X d.x ˝i / H.x/; D I D I Ä k k k N k D NI D NI D N i 1 i 1 D D which contradicts the fact that x S and thus completes the proof. N 2 e following result gives us verifiable necessary and sufficient conditions for the solution uniqueness in (4.37). 162 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Figure 4.5: A generalized Fermat-Torricelli problem with infinite solutions. 3 eorem 4.51 Let i 1˝i . en problem (4.37) has more than one solutions if and only \ D D; 2 if there is a set Œci ; cj ˝k for distinct indices i; j; k 1; 2; 3 that contains a point u R \ 2 f g 2 belonging to the interior of ˝k and such that the index set I.u/ is a singleton. Proof. Let u Œc1; c2 ˝3, u int ˝3 satisfy the conditions in the “if ” part of the theo- 2 \ 2 rem. en u ˝1, u ˝2 and we can choose v u with v Œc1; c2], v ˝1, v ˝2, and … … ¤ 2 … … v int ˝3. is shows that the set I.v/ is a singleton. Let us verify that in this case u is a solu- 2 tion to (4.37). To proceed, we observe first that H.u/ d.u ˝1/ d.u ˝2/ d.u ˝3/ c1 c2 r1 r2: D I C I C I D k k Choose ı > 0 such that IB.u ı/ ˝1 , IB.u ı/ ˝2 , and IB.u ı/ ˝3. For every I \ D; I \ D; I x IB.u ı/, we get 2 I H.x/ d.x ˝2/ d.x ˝3/ x c1 x c2 r1 r2 c1 c2 r1 r2 H.u/; D I C I D k k C k k k k D which implies that u is a local (and hence global) minimizer of H. Similar arguments show that v S, so the solution set S is not a singleton. 2 To justify the “only if ” part, suppose that S is not a singleton and pick two distinct elements x; y S, which yields Œx; y S by Proposition 4.45. If there is a solution to (4.37) not belonging N N 2 N N to ˝i for every i 1; 2; 3 , then S reduces to a singleton due to Proposition 4.47, a contradiction. 2 f g us we can assume by the above that ˝1 contains infinitely many solutions. en its interior int ˝1 also contains infinitely many solutions by the obvious strict convexity of the ball ˝1. If there is such a solution u with I.u/ 1 , then u int ˝1 and u ˝2 as well as u ˝3. is D f g 2 … … implies that u Œc2; c3, since the opposite ensures the solution uniqueness by Proposition 4.49. 2 Consider the case where the set I.u/ contains two elements for every solution belonging to int ˝1. en there are infinitely many solutions belonging to the intersection of these two sets, which is strictly convex in this case. Hence there is a solution to (4.37) that belongs to the interior of this 4.5. A GENERALIZED FERMAT-TORRICELLI PROBLEM 163 intersection, which contradicts Proposition 4.50 and thus completes the proof of the theorem. Let us now consider yet another particular case of problem (4.34), where the constant dy- namics F is given by the closed unit ball. It is formulated as m minimize D.x/ X d.x ˝i / subject to x ˝0: (4.41) WD I 2 i 1 D For x Rn, define the index subsets of 1; : : : ; m by 2 f g I.x/ ˚i 1; : : : ; m ˇ x ˝i « and J.x/ ˚i 1; : : : ; m ˇ x ˝i «: (4.42) WD 2 f g ˇ 2 WD 2 f g ˇ … It is obvious that I.x/ J.x/ 1; : : : ; m and I.x/ J.x/ for all x Rn. e next theo- [ D f g \ D; 2 rem fully characterizes optimal solutions to (4.41) via projections to ˝i . eorem 4.52 e following assertions hold for problem (4.41): (i) Let x ˝0 be a solution to (4.41). For ai .x/ Ai .x/ as i J.x/, we have N 2 N Á N 2 N X ai .x/ X Ai .x/ N.x ˝0/; (4.43) N 2 N C NI i J.x/ i I.x/ 2 N 2 N where each set Ai .x/, i 1; : : : ; m, is calculated by N D 8 x ˘.x ˝i / N NI for x ˝i ; ˆ d.x ˝i / N … Ai .x/ < NI (4.44) N D ˆ N.x ˝i / IB for x ˝i : : NI \ N 2 (ii) Conversely, if x ˝0 satisfies (4.43), then x is a solution to (4.41). N 2 N Proof. To verify (i), fix an optimal solution x to problem (4.41) and equivalently describe it as an N optimal solution to the unconstrained optimization problem: minimize D.x/ ı.x ˝/; x Rn : (4.45) C I 2 Applying the subdifferential Fermat rule to (4.45), we characterize x by N n 0 @ X d. ˝i / ı. ˝/Á.x/: (4.46) 2 I C I N i 1 D 164 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Since all of the functions d. ˝i / for i 1; : : : ; n are convex and continuous, we employ the I D subdifferential sum rule of eorem 2.15 to (4.46) and arrive at 0 X @d.x ˝i / X d.x ˝i / N.x ˝0/ 2 NI C NI C NI i J.x/ i I.x/ 2 N 2 N X Ai .x/ X Ai .x/ N.x ˝0/ D N C N C NI i J.x/ i I.x/ 2 N 2 N X ai .x/ X Ai .x/ N.x ˝0/; D N C N C NI i J.x/ i I.x/ 2 N 2 N which justifies the claimed inclusion X ai .x/ X Ai .x/ N.x ˝0/: N 2 N C NI i J.x/ i I.x/ 2 N 2 N e converse assertion (ii) is verified similarly by taking into account the “if and only if ” property of the proof of (i). Let us specifically examine problem (4.41) in the case where the intersections of the con- straint set ˝0 with the target sets ˝i are empty. Corollary 4.53 Consider problem (4.41) with ˝0 ˝i for i 1; : : : ; m. Given x ˝0, de- \ D; D N 2 fine the elements x ˘.x ˝i / ai .x/ N NI 0; i 1; : : : ; m: (4.47) N WD d.x ˝i / ¤ D NI en x ˝0 is an optimal solution to (4.41) if and only if N 2 m X ai .x/ N.x ˝0/: (4.48) N 2 NI i 1 D Proof. In this case we have that x ˝i for all i 1; : : : ; m, which means that I.x/ and N … D N D; J.x/ 1; : : : ; m . us our statement is a direct consequence of eorem 4.52. N D f g For more detailed solution of (4.41), consider its special case of m 3. D eorem 4.54 Let m 3 in the framework of eorem 4.52, where the target sets ˝1; ˝2, and D n ˝3 are pairwise disjoint, and where ˝0 R . e following hold for the optimal solution x to D N (4.41) with the sets ai .x/ defined by (4.47): N (i) If x belongs to one of the sets ˝i , say ˝1, then we have N ˝a2; a3 1=2 and a2 a3 N.x ˝1/: i Ä 2 NI 4.5. A GENERALIZED FERMAT-TORRICELLI PROBLEM 165 (ii) If x does not belong to any of the three sets ˝1, ˝2, and ˝3, then N ai ; aj 1=2 for i j as i; j ˚1; 2; 3«: h i D ¤ 2 Conversely, if x satisfies either (i) or (ii), then it is an optimal solution to (4.41). N Proof. To verify (i), we have from eorem 4.52 that x ˝1 solves (4.41) if and only if N 2 0 @d.x ˝1/ @d.x ˝2/ @d.x ˝3/ 2 NI C NI C NI @d.x ˝1/ a2 a3 N.x ˝1/ IB a2 a3: D NI C C D NI \ C C us a2 a3 N.x ˝1/ IB, which is equivalent to the validity of a2 a3 IB and a2 2 NI \ 2 a3 N.x ˝1/. By Lemma 4.31, this is also equivalent to 2 NI ˝a2; a3 1=2 and a2 a3 N.x ˝1/; i Ä 2 NI which gives (i). To justify (ii), we similarly conclude that in the case where x ˝i for i 1; 2; 3, N … D this element solves (4.41) if and only if 0 @d.x ˝1/ @d.x ˝2/ @d.x ˝3/ a1 a2 a3: 2 NI C NI C NI D C C en Lemma 4.32 tells us that the latter is equivalent to ai ; aj 1=2 for i j as i; j ˚1; 2; 3«: h i D ¤ 2 Conversely, the validity of either (i) or (ii) and the convexity of ˝i ensure that 0 @d.x ˝1/ @d.x ˝2/ @d.x ˝3/ @D.x/; 2 NI C NI C NI D N and thus x is an optimal solution to the problem under consideration. N Figure 4.6: Generalized Fermat-Torricelli problem for three disks. 2 Example 4.55 Let the sets ˝1, ˝2, and ˝3 in problem (4.37) be closed balls in R of radius r 1 centered at the points .0; 2/, . 2; 0/, and .2; 0/, respectively. We can easily see that the point D 166 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS .0; 1/ ˝1 satisfies all the conditions in eorem 4.54(i), and hence it is an optimal solution (in 2 fact the unique one) to this problem. More generally, consider problem (4.37) in R2 generated by three pairwise disjoint disks denoted by ˝i , i 1; 2; 3. Let c1, c2, and c3 be the centers of the disks. Assume first that either D 2 the line segment Œc2; c3 R intersects ˝1, or Œc1; c3 intersects ˝2, or Œc1; c2 intersects ˝3. It 2 is not hard to check that any point of the intersections (say, of the sets ˝1 and Œc2; c3 for defi- niteness) is an optimal solution to the problem under consideration, since it satisfies the necessary and sufficient optimality conditions of eorem 4.54(i). Indeed, if x is such a point, then a2 and N a3 are unit vectors with a2; a3 1 and a2 a3 0 N.x ˝1/. If the above intersection h i D D 2 NI assumptions are violated, we define the three points q1; q2, and q3 as follows. Let u and v be in- tersection points of Œc1; c2 and Œc1; c3 with the boundary of the disk centered at c1, respectively. en we see that there is a unique point q1 on the minor curve generated by u and v such that the measures of angle c1q1c2 and c1q1c3 are equal. e points q2 and q3 are defined similarly. eorem 4.54 yields that whenever the angle c2q1c3, or c1q2c3, or c2q3c1 equals or exceeds 120ı (say, the angle c2q1c3 does), then the point x q1 is an optimal solution to the problem under N WD consideration. Indeed, in this case a2 and a3 from (4.47) are unit vectors with a2; a3 1=2 h i Ä and a2 a3 N.x ˝1/ because the vector a2 a3 is orthogonal to ˝1. 2 NI If none of these angles equals or exceeds 120ı, there is a point q not belonging to ˝i as i 1; 2; 3 such that the angles c1qc2 c2qc3 c3qc1 are of 120 and that q is an optimal D D D ı solution to the problem under consideration. Observe that in this case the point q is also a unique optimal solution to the classical Fermat-Torricelli problem determined by the points c1; c2, and c3; see Figure 4.6. Finally in this section, we discuss the application of the subgradient algorithm from Sec- tion 4.3 to the numerical solution of the generalized Fermat-Torricelli problem (4.34). eorem 4.56 Consider the generalized Fermat-Torricelli problem (4.34) and assume that S ¤ for its solution set. Picking a sequence ˛k of positive numbers and a starting point x1 ˝0, ; f g 2 form the iterative sequence m xk 1 ˘xk ˛k X vik ˝0Á; k 1; 2; : : : ; (4.49) C WD I D i 1 D with an arbitrary choice of subgradient elements vik @F .!ik xk/ N.!ik ˝i / for some !ik ˘F .xk ˝i / if xk ˝i (4.50) 2 \ I 2 I … and vik 0 otherwise, where ˘F .x ˝/ stands for the generalized projection operator (3.21), WD I and where F is the Minkowski gauge (3.11). Define the value sequence Vk min ˚H.xj / ˇ j 1; : : : ; k«: WD ˇ D 4.5. A GENERALIZED FERMAT-TORRICELLI PROBLEM 167 (i) If the sequence ˛k satisfies conditions (4.13), then Vk converges to the optimal value V of f g f g problem (4.34). Furthermore, we have the estimate 2 2 2 d.x1 S/ ` Pk1 1 ˛k 0 Vk V I C D ; Ä Ä k 2 Pi 1 ˛k D where ` 0 is a Lipschitz constant of cost function H. / on Rn. (ii) If the sequence ˛k satisfies conditions (4.15), then Vk converges to the optimal value V f g f g and xk in (4.49) converges to an optimal solution x S of problem (4.34). f g N 2 Proof. is is a direct application of the projected subgradient algorithm described in Proposi- tion 4.28 to the constrained optimization problem (4.34) by taking into account the results for the subgradient method in convex optimization presented in Section 4.3. Let us specify the above algorithm for the case of the unit ball F IB in (4.34). D n Corollary 4.57 Let F IB R in (4.34), let ˛k be a sequence of positive numbers, and let D f g x1 ˝0 be a starting point. Consider algorithm (4.49) with @F . / calculated by 2 8 ˘.xk ˝i / xk I if xk ˝i ; ˆ d.xk ˝i / … @F .!ki xk/ < I D ˆ 0; if xk ˝i : : 2 en all the conclusions of eorem 4.56 hold for this specification. Proof. Immediately follows from eorem 4.56 with F .x/ x . D k k Next we illustrate applications of the subgradient algorithm of eorem 4.56 to solving the generalized Fermat-Torricelli problem (4.34) formulated via the minimal time function with non-ball dynamics. Note that to implement the subgradient algorithm (4.49), it is sufficient to calculate just one subgradient from the set on the right-hand side of (4.50) for each target set. In the following example we consider for definiteness the dynamics F given by the square Œ 1; 1 Œ 1; 1 in the plane. In this case the corresponding Minkowski gauge (3.11) is given by the formula 2 F .x1; x2/ max ˚ x1 ; x2 «; .x1; x2/ R : (4.51) D j j j j 2 Example 4.58 Consider the unconstrained generalized Fermat-Torricelli problem (4.34) with the dynamics F Œ 1; 1 Œ 1; 1 and the square targets ˝i of right position (with sides parallel D to the axes) centered at .ai ; bi / with radii ri (half of the side) as i 1; : : : ; m. Given a sequence D 168 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Figure 4.7: Minimal time function with square dynamic of positive numbers ˛k and a starting point x1, construct the iterations xk .x1k; x2k/ by f g D algorithm (4.49). By eorem 3.39, subgradients vik can be chosen by 8 .1; 0/ if x2k bi x1k ai and x1k > ai ri ; ˆ j j Ä C ˆ ˆ ˆ . 1; 0/ if x2k bi ai x1k and x1k < ai ri ; ˆ j j Ä ˆ ˆ vik < .0; 1/ if x1k ai x2k bi and x2k > bi ri ; D j j Ä C ˆ ˆ .0; 1/ x a b x x < b r ; ˆ if 1k i i 2k and 2k i i ˆ j j Ä ˆ ˆ :ˆ .0; 0/ otherwise. 4.6 A GENERALIZED SYLVESTER PROBLEM In [31] James Joseph Sylvester proposed the smallest enclosing circle problem: Given a finite number of points in the plane, find the smallest circle that encloses all of the points. After more than a century, location problems of this type remain active as they are mathematically beautiful and have meaningful real-world applications. is section is devoted to the study by means of convex analysis of the following generalized version of the Sylvester problem formulated in [18]; see also [19, 20] for further developments. Let F Rn be a closed, bounded, convex set containing the origin as an interior point. Define the generalized ball centered at x Rn with radius r > 0 by 2 GF .x r/ x rF: I WD C 4.6. A GENERALIZED SYLVESTER PROBLEM 169 It is obvious that for F IB the generalized ball GF .x r/ reduces to the closed ball of radius D I r centered at x. Given now finitely many nonempty, closed, convex sets ˝i for i 0; : : : ; m, D the generalized Sylvester problem is formulated as follows: Find a point x ˝0 and the smallest 2 number r 0 such that GF .x r/ ˝i for all i 1; : : : ; m: I \ ¤ ; D 2 2 Observe that when ˝0 R and all the sets ˝i for i 1; : : : ; m are singletons in R , the gen- D D eralized Sylvester problem becomes the classical Sylvester enclosing circle problem. e sets ˝i for i 1; : : : ; m are often called the target sets. D To study the generalized Sylvester problem, we introduce the cost function T .x/ max ˚TF .x ˝i / ˇ i 1; : : : ; m« WD I ˇ D via the minimal time function (3.9) and consider the following optimization problem: minimize T .x/ subject to x ˝0: (4.52) 2 We can show that the generalized ball GF .x r/ with x ˝0 is an optimal solution of the gener- I N 2 alized Sylvester problem if and only if x is an optimal solution of the optimization problem (4.52) N with r T .x/. D N Our first result gives sufficient conditions for the existence of solutions to (4.52). Proposition 4.59 e generalized Sylvester problem (4.52) admits an optimal solution if the set m ˝0 Ti 1.˝i ˛F / is bounded for all ˛ 0. is holds, in particular, if there exists an index \ D i0 0; : : : ; m such that the set ˝i is compact. 2 f g 0 Proof. For any ˛ 0, consider the level set L˛ ˚x ˝0 ˇ T .x/ ˛« WD 2 ˇ Ä and then show that in fact we have m L˛ ˝0 Œ\.˝i ˛F /: (4.53) D \ i 1 D Indeed, it holds for any nonempty, closed, convex set Q that TF .x Q/ ˛ if and only if .x I Ä C ˛F / Q , which is equivalent to x Q ˛F . If x L˛, then x ˝0 and TF .x ˝i / ˛ \ ¤ ; 2 2 2 I Ä for i 1; : : : ; m. us x ˝i ˛F for i 1; : : : ; m, which verifies the inclusion “ ” in (4.53). D 2 D e proof of the opposite inclusion is also straightforward. It is clear furthermore that the bound- m edness of the set ˝0 T .˝i ˛F / for all ˛ 0 ensures that the level sets for the cost \ i 1 function in (4.52) are bounded.D us it follows from eorem 4.11 that problem (4.52) has an optimal solution. e last statement is obvious. 170 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Now we continue with the study of the uniqueness of optimal solutions to (4.52) splitting the proof into several propositions. Proposition 4.60 Consider a convex function g.x/ 0 on a nonempty, convex set ˝ Rn. en the function defined by h.x/ .g.x//2 is strictly convex on ˝ if and only if g is not constant on any WD line segment Œa; b ˝ with a b. ¤ Proof. Suppose that h is strictly convex on ˝ and suppose on the contrary that g is constant on a line segment Œa; b with a b, then h is also constant on this line segment, so it cannot be ¤ strictly convex. Conversely, suppose g is not constant on any segment Œa; b ˝ with a b. ¤ Observe that h is a convex function on ˝ since it is a composition of a quadratic function, which is a nondecreasing convex function on Œ0; /, and a convex function whose range is a subset 1 of Œ0; /. Let us show that it is strictly convex on ˝. Suppose by contradiction that there are 1 t .0; 1/ and x; y ˝, x y, with 2 2 ¤ h.z/ th.x/ .1 t/h.y/ with z tx .1 t/y: D C WD C en .g.z//2 t.g.x//2 .1 t/.g.y//2. Since g.z/ tg.x/ .1 t/g.y/, one has D C Ä C 2 2 2 g.z/ t 2 g.x/ .1 t/2 g.y/ 2t.1 t/g.x/g.y/; Ä C C which readily implies that 2 2 2 2 t g.x/ .1 t/ g.y/ t 2 g.x/ .1 t/2 g.y/ 2t.1 t/g.x/g.y/: C Ä C C us .g.x/ g.y//2 0, so g.x/ g.y/. is shows that h.x/ h.z/ h.y/ for the point Ä D D D z .x; y/ defined above. We arrive at a contradiction by showing that h is constant on the line 2 segment Œz; y. Indeed, fix any u .z; y/ and observe that 2 h.u/ h.z/ .1 /h.y/ h.z/ for some .0; 1/: Ä C D 2 On the other hand, since z lies between x and u, we have h.z/ h.x/ .1 /h.u/ h.z/ .1 /h.z/ h.z/ for some .0; 1/: Ä C Ä C D 2 It gives us h.z/ h.x/ .1 /h.u/ h.z/ .1 /h.u/, and hence h.u/ h.z/. is D C D C D contradicts the assumption that g.u/ ph.u/ is not constant on any line segment Œa; b with D a b and thus completes the proof of the proposition. ¤ Proposition 4.61 Let both sets F and ˝ in Rn be nonempty, closed, strictly convex, where F contains the origin as an interior point. en the convex function TF . ˝/ is not constant on any line segment I Œa; b Rn such that a b and Œa; b ˝ . ¤ \ D; 4.6. A GENERALIZED SYLVESTER PROBLEM 171 Proof. To prove the statement of the proposition, suppose on the contrary that there is Œa; b n R with a b such that Œa; b ˝ and TF .x ˝/ r for all x Œa; b. Choosing u ¤ \ D; I D 2 2 ˘F .a ˝/ and v ˘F .b ˝/, we get F .u a/ F .v b/ r. Let us verify that u v. In- I 2 I D D ¤ deed, the opposite gives us F .u a/ F .u b/ r, which implies by r > 0 that D D u a u b F Á F Á 1; r D r D u a u b and thus bd F and bd F . Since F is strictly convex, it yields r 2 r 2 1 u a 1 u b 1 a b Á Á u C Á int F: 2 r C 2 r D r 2 2 Hence we arrive at a contradiction by having a b a b TF C ˝Á F u C Á < r: 2 I Ä 2 To finish the proof of the proposition, fix t .0; 1/ and get 2 r TF ta .1 t/b ˝ F tu .1 t/v .ta .1 t/b/ D C I Ä C C tF .u a/ .1 t/F .v b/ r; Ä C D which implies that tu .1 t/v ˘F .ta .1 t/b ˝/. us tu .1 t/v bd ˝, which C 2 C I C 2 contradicts the strict convexity of ˝ and completes the proof. Figure 4.8: Minimal time function for strictly convex sets. n Proposition 4.62 Let ˝ R be a nonempty, closed, convex set and let hi ˝ R be nonnegative, W ! continuous, convex functions for i 1; : : : ; m. Define D .x/ max ˚h1.x/; : : : ; hm.x/« WD 172 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS and suppose that .x/ r > 0 on Œa; b for some line segment Œa; b ˝ with a b. en there D ¤ exists a line segment Œ˛; ˇ Œa; b with ˛ ˇ and an index i0 1; : : : ; m such that we have ¤ 2 f g hi .x/ r for all x Œ˛; ˇ. 0 D 2 Proof. ere is nothing to prove for m 1. When m 2, consider the function D D .x/ max ˚h1.x/; h2.x/«: WD en the conclusion is obvious if h1.x/ r for all x Œa; b. Otherwise we find x0 Œa; b with D 2 2 h1.x0/ < r and a subinterval Œ˛; ˇ Œa; b, ˛ ˇ, such that ¤ h1.x/ < r for all x Œ˛; ˇ: 2 us h2.x/ r on this subinterval. To proceed further by induction, we assume that the conclu- D sion holds for a positive integer m 2 and show that the conclusion holds for m 1 functions. C Denote .x/ max ˚h1.x/; : : : ; hm.x/; hm 1.x/«: WD C en we have .x/ max h1.x/; k1.x/ with k1.x/ max h2.x/; : : : ; hm 1.x/ , and the con- D f g WD f C g clusion follows from the case of two functions and the induction hypothesis. Now we are ready to study the uniqueness of an optimal solution to the generalized Sylvester problem (4.52). eorem 4.63 In the setting of problem (4.52) suppose further that the sets F and ˝i for i 1; : : : ; m are strictly convex. en problem (4.52) has at most one optimal solution if and D m only if i 0˝i contains at most one element. \ D Proof. Define K.x/ .T .x//2 and consider the optimization problem WD minimize K.x/ subject to x ˝0: (4.54) 2 It is obvious that x ˝0 is an optimal solution to (4.52) if and only if it is an optimal solution N 2 m K to problem (4.54). We are going to prove that if i 0˝i contains at most one element, then is \ D strictly convex on ˝0, and hence problem (4.52) has at most one optimal solution. Indeed, suppose that K is not strictly convex on ˝0. By Proposition 4.60, we find a line segment Œa; b ˝0 with a b and a number r 0 so that ¤ T .x/ max ˚TF .x ˝i / ˇ i 1; : : : ; m« r for all x Œa; b: D I ˇ D D 2 m It is clear that r > 0, since otherwise Œa; b T ˝i , which contradicts the assumption that i 0 this set contains at most one element. PropositionD 4.62 tells us that there exist a line segment Œ˛; ˇ Œa; b with ˛ ˇ and an index i0 1; : : : ; m such that ¤ 2 f g TF .x ˝i / r for all x Œ˛; ˇ: I 0 D 2 4.6. A GENERALIZED SYLVESTER PROBLEM 173 Since r > 0, we have x ˝i0 for all x Œ˛; ˇ, which contradicts Proposition 4.61. … 2 m Conversely, let problem (4.52) have at most one solution. Suppose now that the set i 0˝i m \ D contains more than one element. It is clear that Ti 0 ˝i is the solution set of (4.52) with the op- timal value equal to zero. us this problem has moreD than one solution, which is a contradiction that completes the proof of the theorem. e following corollary gives a sufficient condition for the uniqueness of optimal solutions to the generalized Sylvester problem (4.52). Corollary 4.64 In the setting of (4.52) suppose that the sets F and ˝i for i 1; : : : ; m are strictly m D convex. If T ˝i contains at most one element and if one of the sets among ˝i for i 0; : : : ; m is i 0 D bounded, thenD problem (4.52) has a unique optimal solution. m Proof. Since Ti 0 ˝i contains at most one element, problem (4.52) has at most one optimal solution by eoremD 4.63. us it has exactly one solution because the solution set is nonempty by Proposition 4.59. Next we consider a special unconstrained case of (4.52) given by n minimize M.x/ max ˚d.x ˝i / ˇ i 1; : : : ; m« subject to x R ; (4.55) WD I ˇ D 2 n for nonempty, closed, convex sets ˝i under the assumption that ˝i . Problem (4.55) \i 1 D; corresponds to (4.52) with F IB Rn. Denote the set of active indicesD for (4.55) by D n I.x/ ˚i 1; : : : ; m ˇ M.x/ d.x ˝i /«; x R : WD 2 f g ˇ D I 2 Further, for each x ˝i , define the singleton … x ˘.x ˝i / ai .x/ I (4.56) WD d.x ˝i / I where the projection ˘.x ˝i / is unique. We have M.x/ > 0 as i I.x/ due to the assumption m I 2 Ti 1 ˝i . is ensures that x ˝i automatically for any i I.x/. D D; … 2 Let us now present necessary and sufficient optimality conditions for (4.55) based on the gen- eral results of convex analysis and optimization developed above. eorem 4.65 e element x Rn is an optimal solution to problem (4.55) under the assump- N 2 tions made if and only if we have the inclusion x co ˚˘.x ˝i / ˇ i I.x/« (4.57) N 2 NI ˇ 2 N In particular, for the case where ˝i ai , i 1; : : : ; m, problem (4.55) has a unique solution D f g D and x is the problem generated by ai if and only if N x co ˚ai ˇ i I.x/«: (4.58) N 2 ˇ 2 N 174 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Proof. It follows from Proposition 2.35 and Proposition 2.54 that the condition 0 co [ @d.x ˝i / 2 NI i I.x/ 2 N is necessary and sufficient for optimality of x in (4.55). Employing now the distance function N subdifferentiation from eorem 2.39 gives us the minimum characterization 0 co ˚ai .x/ ˇ i I.x/«; (4.59) 2 N ˇ 2 N via the singletons ai .x/ calculated in (4.56). It is easy to observe that (4.59) holds if and only if N there are i 0 for i I.x/ such that Pi I.x/ i 1 and 2 N 2 N D x !i 0 X i N N with !i ˘.x ˝i /: D M.x/ WD NI i I.x/ 2 N N is equation is clearly equivalent to 0 X i .x !i /; or x X i !i ; D N N N D N i I.x/ i I.x/ 2 N 2 N which in turn is equivalent to inclusion (4.57). Finally, for ˝i ai as i 1; : : : ; m, the latter D f g D inclusion obviously reduces to condition (4.58). In this case the problem has a unique optimal solution by Corollary 4.64. To formulate the following result, we say that the ball IB.x r/ touches the set ˝i if the in- NI tersection set ˝i IB.x r/ is a singleton. \ NI Corollary 4.66 Consider problem (4.55) under the assumptions above. en any smallest intersecting ball touches at least two target sets among ˝i , i 1; : : : ; m. D Proof. Let us first verify that there are at least two indices in I.x/ provided that x is a solution to N N (4.55). Suppose the contrary, i.e., I.x/ i0 . en it follows from (4.57) that N D f g x ˘.x ˝i / and d.x ˝i / M.x/ > 0; N 2 NI 0 NI 0 D N a contradiction. Next we show that IB.x r/ with r M.x/ touches ˝i when i I.x/. Indeed, NI WD N 2 N if on the contrary there are u; v ˝i such that 2 u; v IB.x r/ ˝i with u v; 2 NI \ ¤ en, by taking into account that d.x ˝i / r, we get NI D u x r d.x ˝i / and v x r d.x ˝i /; k Nk Ä D NI k Nk Ä D NI so u; v ˘.x ˝i /. is contradicts the fact that the projector ˘.x ˝i / is a singleton and thus 2 NI NI completes the proof of the corollary. 4.6. A GENERALIZED SYLVESTER PROBLEM 175 Now we illustrate the results obtained by solving the following two examples. 2 Example 4.67 Let x solve the smallest enclosing ball problem generated by ai R , i 1; 2; 3. N 2 D Corollary 4.66 tells us that there are either two or three active indices. If I.x/ consists of N two indices, say I.x/ 2; 3 , then x co a2; a3 and x a2 x a3 > x a1 by N D f g N 2 af2 a3g k N k D k N k k N k eorem 4.65. In this case we have x C and a2 a1; a3 a1 < 0. If conversely N D 2 h i a2 a1; a3 a1 < 0, then the angle of the triangle formed by a1, a2, and a3 at the vertex h i a1 is obtuse; we include here the case where all the three points ai are on a straight line. It is a2 a3 a2 a3 easy to see that I C Á 2; 3 and that the point x C satisfies the assumption 2 D f g N D 2 of eorem 4.65 thus solving the problem. Observe that in this case the solution of (4.55) is the midpoint of the side opposite to the obtuse vertex. If none of the angles of the triangle formed by a1; a2; a3 is obtuse, then the active index set I.x/ consists of three elements. In this case x is the unique point satisfying N N x co ˚a1; a2; a3« and x a1 x a2 x a3 : N 2 k N k D k N k D k N k In the other words, x is the center of the circumscribing circle of the triangle. N Figure 4.9: Smallest intersecting ball problem for three disks. 2 Example 4.68 Let ˝i IB.ci ; ri / with ri > 0 and i 1; 2; 3 be disjoint disks in R , and let x D D N be the unique solution of the corresponding problem (4.55). Consider first the case where one of the line segments connecting two of the centers intersects the other disks, e.g., the line segment connecting c2 and c3 intersects ˝1. Denote u2 c2c3 bd ˝2 and u3 c2c3 bd ˝3, and let WD \ WD \ x be the midpoint of u2u3. en I.x/ 2; 3 and we can apply eorem 4.65 to see that x is N N D f g N the solution of the problem. 176 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS It remains to consider the case where any line segment connecting two centers of the disks does not intersect the remaining disk. In this case denote u1 c1c2 bd ˝1; v1 c1c3 bd ˝1; WD \ WD \ u2 c2c3 bd ˝2; v2 c2c1 bd ˝2; WD \ WD \ u3 c3c1 bd ˝3; v3 c3c2 bd ˝3; WD \ WD \ a1 c1m1 bd ˝1; a2 c2m2 bd ˝2; a3 c3m3 bd ˝3; WD \ D \ WD \ where m1 is the midpoint of u2v3, m2 is the midpoint of u3v1, and m3 is the midpoint of u1v2. Suppose that one of the angles u2a1v3, u3a2v1, u1a3v2 is greater than or equal to 90ı; e.g., u2a1v3 is greater than 90 . en I.m1/ 2; 3 , and thus x m1 is the unique solution of this ı D f g N D problem by eorem 4.65. If now2 we suppose2 that all2 of the aforementioned angles are less than or equal to 90 , then I.x/ 3 and the smallest disk we are looking for is the unique disk that 2 ı N D touches three other disks. e construction of this disk is the celebrated problem of Apollonius. Finally in this section, we present and illustrate the subgradient algorithm to solve numeri- cally the generalized Sylvester problem (4.52). eorem 4.69 In the setting of problem (4.52) suppose that the solution set S of the problem is nonempty. Fix x1 ˝0 and define the sequences of iterates by 2 xk 1 ˘ xk ˛kvk ˝0; k N; C WD I 2 where ˛k is a sequence of positive numbers, and where f g 0 if xk ˝i ; vk f g 2 2 @F .!k xk/ N.!k ˝i / if xk ˝i ; \ I … where !k ˘F .xk ˝i / for some i I.xk/. Define the value sequence 2 I 2 Vk min ˚T .xj / ˇ j 1; : : : ; k«: WD ˇ D (i) If the sequence ˛k satisfies conditions (4.13), then Vk converges to the optimal value V of f g f g the problem. Furthermore, we have the estimate 2 2 2 d.x1 S/ ` Pk1 1 ˛k 0 Vk V I C D ; Ä Ä k 2 Pi 1 ˛k D where ` 0 is a Lipschitz constant of cost function T . / on Rn. (ii) If the sequence ˛k satisfies conditions (4.15), then Vk converges to the optimal value V f g f g and xk in (4.49) converges to an optimal solution x S of the problem. f g N 2 Proof. Follows from the results of Section 4.3 applied to problem (4.52). 4.6. A GENERALIZED SYLVESTER PROBLEM 177 e next examples illustrate the application of the subgradient algorithm of eorem 4.69 in particular situations. Example 4.70 Consider problem (4.55) in R2 with m target sets given by the squares ˝i S.!i ri / !1i ri ;!1i ri !2i ri ;!2i ri ; i 1; : : : ; m: D I WD C C D Denote the vertices of the ith square by v1i .!1i ri ;!2i ri /; v2i .!1i ri ;!2i WD C C WD C ri /; v3i .!1i ri ;!2i ri /; v4i .!1i ri ;!2i ri /, and let xk .x1k; x2k/. Fix an in- WD WD C WD dex i I.xk/ and then choose the elements vk from eorem 4.69 by 2 xk v1i 8 if x1k !1i > ri and x2k !2i > ri ; ˆ xk v1i ˆ kx v2ik ˆ k x ! < r x ! > r ; ˆ if 1k 1i i and 2k 2i i ˆ xk v2i ˆ kxk v3i k ˆ if x1k !1i < ri and x2k !2i < ri ; ˆ xk v3i ˆ x v vk < k k 4ik D if x1k !1i > ri and x2k !2i < ri ; ˆ xk v4i ˆ .0;k 1/ k x ! r x ! > r ; ˆ if 1k 1i i and 2k 2i i ˆ j j Ä ˆ .0; 1/ if x1k !1i ri and x2k !2i < ri ; ˆ j j Ä ˆ .1; 0/ if x1k !1i > ri and x2k !2i ri ; ˆ j j Ä ˆ . 1; 0/ if x1k !1i < ri and x2k !2i ri : : j j Ä Figure 4.10: Euclidean projection to a square. Now we consider the case of a non-Euclidean norm in the minimal time function. 178 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS 2 Example 4.71 Consider problem (4.52) with F ˚.x1; x2/ R ˇ x1 x2 1« and a WD 2 ˇ j j C j j Ä nonempty, closed, convex target set ˝. e generalized ball GF .x t/ centered at x .x1; x2/ NI N D N N with radius t > 0 has the following diamond shape: 2 GF .x t/ ˚.x1; x2/ R ˇ x1 x1 x2 x2 t«: NI D 2 ˇ j N j C j N j Ä e distance from x to ˝ and the corresponding projection are given by N TF .x ˝/ min ˚t 0 ˇ GF .x t/ ˝ «; (4.60) NI D ˇ NI \ ¤ ; ˘F .x ˝/ GF .x t/ ˝ with t TF .x ˝/: NI D NI \ D NI Let ˝i be the squares S.!i ri /, i 1; : : : ; m, given in Example 4.70. en problem (4.55) can be I D interpreted as follows: Find a set of the diamond shape in R2 that intersects all m given squares. Using the same notation as in Example 4.70, we can see that the elements vk in eorem 4.69 are chosen by 8 .1; 1/ if x1k !1i > ri and x2k !2i > ri ; ˆ ˆ . 1; 1/ if x1k !1i < ri and x2k !2i > ri ; ˆ ˆ . 1; 1/ if x1k !1i < ri and x2k !2i < ri ; ˆ ˆ .1; 1/ if x1k !1i > ri and x2k !2i < ri ; vk < D .0; 1/ if x1k !1i ri and x2k !2i > ri ; ˆ j j Ä ˆ .0; 1/ if x1k !1i ri and x2k !2i < ri ; ˆ j j Ä ˆ .1; 0/ if x1k !1i > ri and x2k !2i ri ; ˆ j j Ä ˆ . 1; 0/ if x1k !1i < ri and x2k !2i ri : : j j Ä 4.7 EXERCISES FOR CHAPTER 4 Exercise 4.1 Given a lower semicontinuous function f Rn R, show that its multiplication W ! .˛f / ˛f.x/ by any scalar ˛ > 0 is lower semicontinuous as well. 7! Exercise 4.2 Given a function f Rn R, show that its lower semicontinuity is equivalent W ! to openess, for every R, of the set 2 n U ˚x R ˇ f .x/ > «: WD 2 ˇ Exercise 4.3 Give an example of two lower semicontinuous real-valued functions whose prod- uct is not lower semicontinuous. 4.7. EXERCISES FOR CHAPTER 4 179 Figure 4.11: Minimal time function with diamond-shape dynamic. Exercise 4.4 Let f Rn R. Prove the following assertions: W ! (i) f is upper semicontinuous (u.s.c.) at x in the sense of Definition 2.89 if and only if f is N lower semicontinuous at this point. (ii) f is u.s.c. at x if and only if lim sup f .xk/ f .x/ for every xk x. N k Ä N !N (iii) f is u.s.c. at x if and only if we have!1 lim sup f .x/ f .x/. N x x Ä N (iv) f is u.s.c. on Rn if and only if the upper level!N set x Rn f .x/ is closed for any real f 2 j g number . (v) f is u.s.c. on Rn if and only if its hypergraphical set (or simply ) hypo f WD .x; / Rn R f .x/ is closed. f 2 j Ä g Exercise 4.5 Show that a function f Rn R is continuous at x if and only if it is both lower W ! N semicontinuous and upper semicontinuous at this point. Exercise 4.6 Prove the equivalence (iii) (iv) in eorem 4.11. ” Exercise 4.7 Consider the real-valued function m 2 n g.x/ X x ai ; x R ; WD k k 2 i 1 D n where ai R for i 1; : : : ; m. Show that g is a convex function and has an absolute minimum 2 D a1 a2 ::: am at the mean point x C C C . N D m 180 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS Exercise 4.8 Consider the problem: minimize x 2 subject to Ax b; k k D where A is an p n matrix with rank A p, and where b Rp. Find the unique optimal solution D 2 of this problem. Exercise 4.9 Give a direct proof of Corollary 4.15 without using eorem 4.14. n Exercise 4.10 Given ai R for i 1; : : : ; m, write MATLAB programs to implement the 2 D subgradient algorithm for minimizing the function f given by: m n (i) f .x/ Pi 1 x ai ; x R . WD mD k k 2 n (ii) f .x/ Pi 1 x ai 1; x R . WD mD k k 2 n (iii) f .x/ Pi 1 x ai ; x R . WD D k k1 2 Exercise 4.11 Write a MATLAB program to implement the projected subgradient algorithm for solving the problem: minimize x 1 subject to Ax b; k k D where A is an p n matrix with rank A p, and where b Rp. D 2 Exercise 4.12 Consider the optimization problem (4.9) and the iterative procedure (4.10) under the standing assumptions imposed in Section 4.3. Suppose that the optimal value V is known and that the sequence ˛k is given by f g 80 if 0 @f.xk/; ˆ 2 ˛k n Exercise 4.13 Let ˝i for i 1; : : : ; m be nonempty, closed, convex subsets of R having a D nonempty intersection. Use the iterative procedure (4.10) for the function f .x/ max ˚d.x ˝i / ˇ i 1; : : : ; m« WD I ˇ D m with the sequence ˛k given in (4.61) to derive an algorithm for finding x Ti 1 ˝i . f g N 2 D Exercise 4.14 Derive the result of Corollary 4.20 from condition (ii) of eorem 4.14. 4.7. EXERCISES FOR CHAPTER 4 181 Exercise 4.15 Formulate and prove a counterpart of eorem 4.27 for the projected subgradient algorithm (4.20) to solve the constrained optimization problem (4.19). Exercise 4.16 Prove the converse implication in Lemma 4.31. Exercise 4.17 Write a MATLAB program to implement the Weiszfeld algorithm. Exercise 4.18 Let ai R for i 1; : : : ; m. Solve the problem 2 D m minimize X x ai ; x R : j j 2 i 1 D Exercise 4.19 Give a simplified proof of eorem 4.51. Exercise 4.20 Prove assertion (ii) of eorem 4.52. Exercise 4.21 Give a complete proof of eorem 4.56. Exercise 4.22 Let ˝i Œai ; bi R for i 1; : : : ; m be m disjoint intervals. Solve the fol- WD 2 D lowing problem: m minimize f .x/ X d.x ˝i /; x R : WD I 2 i 1 D Exercise 4.23 Write a MATLAB program to implement the subgradient algorithm for solving the problem: minimize f .x/; x Rn; 2 m where f .x/ Pi 1 d.x ˝i / with ˝i IB.ci ri / for i 1; : : : ; m. WD D I WD I D Exercise 4.24 Complete the solution of the problem in Example 4.58. Write a MATLAB program to implement the algorithm. Exercise 4.25 Give a simplified proof of eorem 4.63 in the case where F is the closed unit ball of Rn and the target sets are closed Euclidean balls of Rn. Exercise 4.26 Give a simplified proof of eorem 4.63 in the case where F is the closed unit ball of Rn. Exercise 4.27 Write a MATLAB program to implement the subgradient algorithm for solving the problem: minimize f .x/; x Rn; 2 182 4. APPLICATIONS TO OPTIMIZATION AND LOCATION PROBLEMS where f .x/ max d.x ˝i / i 1; : : : ; m with ˝i IB.ci ri / for i 1; : : : ; m. Note that WD f I j D g WD I D if x is an optimal solution to this problem with the optimal value r, then IB.x r/ is the smallest N NI ball that intersects ˝i for i 1; : : : ; m. D Exercise 4.28 Give a complete proof of eorem 4.69. Exercise 4.29 Complete the solution of the problem in Example 4.70. Write a MATLAB program to implement the algorithm. Exercise 4.30 Complete the solution of the problem in Example 4.71. Write a MATLAB program to implement the algorithm. 183 Solutions and Hints for Selected Exercises is final part of the book contains hints for selected exercises. Rather detailed solutions are given for several problems. EXERCISES FOR CHAPTER 1 Exercise 1.1. Suppose that ˝1 is bounded. Fix a sequence xk ˝1 ˝2 converging to x. f g N For every k N, xk is represented as xk yk zk with yk ˝1, zk ˝2. Since ˝1 is closed 2 D 2 2 and bounded, yk has a subsequence yk` converging to y ˝1. en zk` converges to f g f g N 2 2f g y x ˝2. us x ˝1 ˝2 and ˝ is closed. Let ˝1 .x; y/ R y 1=x; x > 0 N N 2 N 2 2 WD f 2 j g and ˝2 .x; y/ R y 1=x; x > 0 . en ˝1 and ˝2 are nonempty, closed, convex WD f 2 j Ä 2 g sets while ˝1 ˝2 .x; y/ R y > 0« is not closed. D f 2 j Exercise 1.4. (i) Use Exercise 1.3(ii). (ii) We only consider the case where m 2 since the general case is similar. From the D definition of cones, both sets ˝1 and ˝2 contain 0, and so ˝1 ˝2 ˝1 ˝2. en [ C co ˝1 ˝2 ˝1 ˝2. To verify the opposite inclusion, fix any x w1 w2 ˝1 ˝2 f [ g C D C 2 C with w1 ˝1 and w2 ˝2 and then write x Œ.2w1 2w2/=2 co ˝1 ˝2 . 2 2 D C 2 f [ g Exercise 1.7. Compute the first or second derivative of each function. en apply either eorem 1.45 or Corollary 1.46. 2 Exercise 1.8. (i) Use f .x; y/ xy; .x; y/ R . (ii) Take f1.x/ x and f2.x/ x. WD 2 WD WD 2 Exercise 1.9. Let f1.x/ x and f2.x/ x , x R. WD WD 2 Exercise 1.12. Use Proposition 1.39. Note that the function f .x/ xq is convex on Œ0; /. WD 1 Exercise 1.16. Define F Rn Rp by F .x/ K, x Rn. en gph F is a convex subset of W ! WD 2 Rn Rp. e conclusion follows! from Proposition 1.50. 184 SOLUTIONS AND HINTS FOR EXERCISES Exercise 1.18. (i) gph F C . (ii) Consider the function D ( p1 x2 if x 1; f .x/ j j Ä WD otherwise: 1 (iii) Apply Proposition 1.50. Exercise 1.20. (i) Use the definition. (ii) Take f .x/ p x . WD j j m m Exercise 1.22. Suppose that v Pi 1 i vi 0 with Pi 1 i 0. If 0, then C D D C D D ¤ m i Pi 1 1, and so D D m i v X vi aff˚v1; : : : ; vm«; D 2 i 1 D which is a contradiction. us 0, and hence i 0 for i 1; : : : ; m because the elements D D D v1; : : : ; vm are affinely independent. Exercise 1.23. Since m ˝ aff ˝, we have aff m aff ˝. To prove the opposite inclusion, it remains to show that ˝ aff m. By contradiction, suppose that there are v ˝ such that 2 v aff m. It follows from Exercise 1.22 that the vectors v; v1; : : : ; vm are affinely independent … and all of them belong to ˝. is contradicts the condition dim ˝ m. e second equality D can also be proved similarly. Exercise 1.24. (i) We obviously have aff ˝ aff ˝. Observe that the set aff ˝ is closed and contains ˝ since aff ˝ w0 L, where w0 ˝ and L is the linear subspace parallel to aff ˝, D C 2 which is a closed set. us ˝ aff ˝ and the conclusion follows. (ii) Choose ı > 0 so that IB.x ı/ aff ˝ ˝. Since x t.x x/ .1 t/x . t/x is an NI \ N C N D C N C affine combination of some elements in aff ˝ aff ˝, we have x t.x x/ aff ˝. us D N C N 2 u x t.x x/ ˝ when t is small. WD N C N 2 (iii) It follows from (i) that ri ˝ ri ˝. Fix any x ri ˝ and x ri ˝. By (ii) we have 2 N 2 z x t.x x/ ˝ when t > 0 is small, and so x z=.1 t/ tx=.1 t/ .z; x/ ri ˝. WD C N 2 D C C N C 2 N Exercise 1.26. If ˝1 ˝2, then ri ˝1 ri ˝2, and so ri ˝1 ri ˝2 by Exercise 1.24(iii). D D D Exercise 1.27. (i) Fix y B.ri ˝/ and find x ri ˝ such that y B.x/. en pick a vector 2 2 D y ri B.˝/ B.˝/ and take x ˝ with y B.x/. If x x, then y y ri B.˝/. Consider N 2 N 2 N D N DN DN 2 the case where x x. Following the solution of Exercise 1.24(iii), find x ˝ such that x .x; x/ ¤ N 2 2 N and let y B.x/ B.˝/. us y B.x/ .B.x/; B.x// .y; y/e, and so y ri B.˝/e. To WD 2 D 2 N D N 2 completee the proof,e it remains to show that B.˝/ eB.ri ˝/ and thene use Exercise 1.26 to obtain D SOLUTIONS AND HINTS FOR EXERCISES 185 the inclusion ri B.˝/ ri B.ri ˝/ B.ri ˝/: D Since B.ri ˝/ B.˝/, the continuity of B and Proposition 1.73 imply that B.˝/ B.˝/ B.ri ˝/ B.ri ˝/: D us we have B.˝/ B.ri ˝/ and complete the proof. (ii) Consider B.x; y/ x y defined on Rn Rn. WD Exercise 1.28. (i) Let us show that aff .epi f / aff .dom f / R : (4.62) D Fix any .x; / aff .epi f /. en there exist i R and .xi ; i / epi f for i 1; : : : ; m with m 2 2 2 D Pi 1 i 1 such that D D m .x; / X i .xi ; i /: D i 1 D m Since xi dom f for i 1; : : : ; m, we have x Pi 1 i xi aff .dom f /, which yields 2 D D D 2 aff .epi f / aff .dom f / R : To verify the opposite inclusion, fix any .x; / aff .dom f / R and find xi dom f and i m 2 m 2 2 R for i 1; : : : ; m with Pi 1 i 1 and x Pi 1 i xi . Define further ˛i f .xi / R and Dm D D D D WD 2 ˛ Pi 1 i ˛i for which .xi ; ˛i / epi f and .xi ; ˛i 1/ epi f . us WD D 2 C 2 m X i .xi ; ˛i / .x; ˛/ aff .epi f /; D 2 i 1 Dm X i .xi ; ˛i 1/ .x; ˛ 1/ aff .epi f /: C D C 2 i 1 D Letting ˛ 1 gives us WD C .x; / .x; ˛/ .1 /.x; ˛ 1/ aff .epi f /; D C C 2 which justifies the opposite inclusion in (4.62). (ii) See the proof of [9, Proposition 1.1.9]. b a; x Exercise 1.30. (i) ˘.x ˝/ x h ia. I D C a 2 (ii) ˘.x ˝/ x AT .AAT / 1.b kAx/k . I D C (iii) ˘.x ˝/ max x; 0 (componentwise maximum). I D f g 186 SOLUTIONS AND HINTS FOR EXERCISES EXERCISES FOR CHAPTER 2 Exercise 2.1. (i) Apply Proposition 2.1 and the fact that tx ˝ for all t 0. 2 (ii) Use (i) and observe that x ˝ whenever x ˝. 2 2 Exercise 2.2. Observe that the sets ri ˝i for i 1; 2 are nonempty and convex. By eorem 2.5, D there is 0 v Rn such that ¤ 2 v; x v; y for all x ri ˝1; y ri ˝2: h i Ä h i 2 2 Fix x0 ri ˝1 and y0 ri ˝2. It follows from eorem 1.72(ii) that for any x ˝1, y ˝2, 2 2 2 2 and t .0; 1/ we have that x t.x0 x/ ri ˝1 and y t.y0 y/ ri ˝2. en apply the 2 C 2 C 2 inequality above for these elements and let t 0 . ! C Exercise 2.3. We split the solution into four steps. Step 1: If L is a subspace of Rn that contains a nonempty, convex set ˝ with x ˝ and x L, then N … N 2 there is 0 v L such that ¤ 2 sup ˚ v; x ˇ x ˝« < v; x : (4.63) h i ˇ 2 h Ni Proof. By Proposition 2.1, find w Rn such that 2 sup ˚ w; x ˇ x ˝« < w; x : (4.64) h i ˇ 2 h Ni Represent Rn as L L , where L u Rn u; x 0 for all x L . en w u v ˚ ? ? WD f 2 j h i D 2 g D C with u L and v L. Substituting w into (4.64) and taking into account that u; x 0 for 2 ? 2 h i D all x ˝ and that u; x 0, we get (4.63). Note that (4.63) yields v 0. 2 h Ni D ¤ Step 2: Let ˝ Rn be a nonempty, convex set such that 0 ri ˝. en the sets ˝ and 0 can be … f g separated properly. Proof. In the case where 0 ˝, we can separate ˝ and 0 strictly, so the conclusion is obvi- … f g ous. Suppose now that 0 ˝ ri ˝. Let L aff ˝ span ˝, where the latter holds due to 2 n WD D 0 aff ˝, which follows from the fact that ˝ L since L is closed. Repeating the proof of 2 Lemma 2.4, find a sequence xk L with xk 0 and xk ˝ for every k. It follows from f g ! … Step 1 that there is vk L with vk 0 and f g ¤ sup ˚ vk; x ˇ x ˝« < 0: h i ˇ 2 Dividing both sides by vk , we suppose without loss of generality that vk 1 and that vk k k k k D ! v L with v 1. en 2 k k D sup ˚ v; x ˇ x ˝« 0: h i ˇ 2 Ä To finish this proof, it remains to show that there exists x ˝ with v; x < 0. By contradiction, 2 h i suppose that v; x 0 for all x ˝. en v; x 0 on ˝. Since v L aff ˝, we can rep- h im 2 m h i D 2 D resent v as v Pi 1 i wi , where Pi 1 i 1 and wi ˝ for i 1; : : : ; m. is implies that D D D D 2 D SOLUTIONS AND HINTS FOR EXERCISES 187 2 m v v; v Pi 1 i v; wi 0, which is a contradiction. k k D h i D D h i D Step 3: Let ˝ Rn be a nonempty, convex set. If the sets ˝ and 0 can be separated properly, then f g we have 0 ri ˝. … Proof. By definition, find 0 v Rn such that ¤ 2 sup ˚ v; x ˇ x ˝« 0 h i ˇ 2 Ä and then take x ˝ with v; x < 0. Arguing by contradiction, suppose that 0 ri ˝. en N 2 h Ni 2 Exercise 1.24(ii) gives us t > 0 such that 0 t.0 x/ tx ˝. us v; tx 0, which C N D N 2 h Ni Ä yields v; x 0, a contradiction. h Ni Step 4: Justifying the statement of Exercise 2.3. Proof. Define ˝ ˝1 ˝2. By Exercise 1.27(ii) we have ri ˝1 ri ˝2 if and only if WD \ D; 0 ri .˝1 ˝2/ ri ˝1 ri ˝2: … D en the conclusion follows from Step 2 and Step 3. Exercise 2.4. Observe that ˝1 Œ˝2 .0; / for any > 0. \ D; Exercise 2.5. Employing eorem 2.5 and arguments similar to those in Exercise 2.2, we can show that there is 0 v Rn such that ¤ 2 v; x v; y for all x ˝1; y ˝2: h i Ä h i 2 2 By Proposition 2.13, choose 0 v N.x ˝1/ Œ N.x ˝2/ and, using the normal cone ¤ 2 NI \ NI definition, arrive at the conclusion with v1 v, v2 v, ˛1 v1; x , and ˛2 v2; x : WD WD WD h Ni WD h Ni Exercise 2.6. Fix x 0 and v N.x ˝/. en N Ä 2 NI v; x x 0 for all x ˝: h Ni Ä 2 Using this inequality for 0 ˝ and 2x ˝, we get v; x 0. Since ˝ . ; 0 ::: 2 N 2 h Ni D D 1 . ; 0, it follows from Proposition 2.11 that v 0. e opposite inclusion is a direct conse- 1 quence of the definition. Exercise 2.8. Suppose that this holds for any k convex sets with k m and m 2. Consider the Ä sets ˝i for i 1; : : : ; m 1 satisfying the qualification condition D C m 1 C h X vi 0; vi N.x ˝i /i vi 0 for all i 1; : : : ; m 1: D 2 NI H) D D C i 1 D 188 SOLUTIONS AND HINTS FOR EXERCISES Since 0 N.x ˝m 1/, this condition also holds for the first m sets. us 2 NI C m m N x \ ˝i Á X N.x ˝i /: NI D NI i 1 i 1 D D m en apply the induction hypothesis to 1 i 1˝i and 2 ˝m 1. WD \ D WD C Exercise 2.10. Since v @f.x/, we have 2 N v.x x/ f .x/ f .x/ for all x R : N Ä N 2 en pick any x1 < x and use the inequality above. N Exercise 2.12. Define the function (f .x/ if x x; g.x/ f .x/ ı x .x/ N DN WD C f Ng D otherwise 1 via the indicator function of x . en epi g x Œf .x/; /, and hence N..x; g.x// epi g/ n f Ng D f Ng N 1 n N nN I D R . ; 0. It follows from Proposition 2.31 that @g.x/ R and @ı x .x/ R . Employing 1 N D f Ng N D the subdifferential sum rule from Corollary 2.45 gives us Rn @g.x/ @f.x/ Rn; D N D N C which ensures that @f.x/ . N ¤ ; Exercise 2.13. We proceed step by step as in the proof of eorem 2.15. It follows from Exer- cise 1.28 that ri 1 ri ˝1 .0; / and D 1 ri 2 ˚.x; / ˇ x ri ˝1; < v; x x «: D ˇ 2 h Ni It is easy to check, arguing by contradiction, that ri 1 ri 2 . Applying the separation result \ D; from Exercise 2.3, we find 0 .w; / Rn R such that ¤ 2 w; x 1 w; y 2 for all .x; 1/ 1; .y; 2/ 2: h i C Ä h i C 2 2 Moreover, there are .x; 1/ 1 and .y; 2/ 2 satisfying e e 2 e e 2 w; x 1 < w; y 2 : h ei C e h ei C e As in the proof of eorem 2.15, it remains to verify that < 0. By contradiction, assume that 0 and then get D w; x w; y for all x ˝1; y ˝2; h i Ä h i 2 2 w; x < w; y with x ˝1; y ˝2: h ei h ei e 2 e 2 SOLUTIONS AND HINTS FOR EXERCISES 189 us the sets ˝1 and ˝2 are properly separated, and hence it follows from Exercise 2.3 that ri ˝1 ri ˝2 . is is a contradiction, which shows that < 0. To complete the proof, we \ D; only need to repeat the proof of Case 2 in eorem 2.15. Exercise 2.14. Use Exercise 2.13 and proceed as in the proof of eorem 2.44. Exercise 2.16. Use the same procedure as in the proof of eorem 2.51 and the construction of the singular subdifferential to obtain @1.f B/.x/ A @1f .y/ ˚Av ˇ v @1f .y/«: (4.65) ı N D N D ˇ 2 N Alternatively, observe the equality dom .f B/ B 1.dom f /. en apply Proposition 2.23 ı D and Corollary 2.53 to complete the proof. Exercise 2.17. Use the subdifferential formula from eorem 2.61 and then the representation of the composition from Proposition 1.54. n Exercise 2.19. For any v R , we have ˝1 .v/ v and ˝2 .v/ v 1. 2 D k k1 D k k Exercise 2.21. Use first the representations n n .ı˝ /.v/ sup ˚ v; x ı.x ˝/ ˇ x R « sup ˚ v; x ˇ x ˝« ˝ .v/; v R : D h i I ˇ 2 D h i ˇ 2 D 2 Applying then the Fenchel conjugate one more time gives us n .˝ /.x/ sup ˚ x; v ˝ .v/ ˇ v R « ı .x/: D h i ˇ 2 D co ˝ Exercise 2.23. Since f Rn R is convex and positively homogeneous, we have @f.0/ and W ! ¤ ; f .0/ 0. is implies in the case where v ˝ @f.0/ that v; x f .x/ for all x Rn, and D 2 D h i Ä 2 so the equality holds when x 0. us we have D n f .v/ sup ˚ v; x f .x/ ˇ x R « 0: D h i ˇ 2 D If v ˝ f .0/, there is x Rn satisfying v; x > f .x/. It gives us by homogeneity that … D N 2 h Ni N n f .v/ sup ˚ v; x f .x/ ˇ x R « sup ˚ v; tx f .tx/ ˇ t > 0« : D h i ˇ 2 h Ni N ˇ D 1 Exercise 2.26. f . 1 d/ and f .1 d/ . 0 I D 1 0 I D 1 Exercise 2.27. Write t2 t2 x t2d .x t1d/ .1 /x for fixed t1 < t2 < 0: N C D t1 N C C t1 N 190 SOLUTIONS AND HINTS FOR EXERCISES Exercise 2.28. (i) For d Rn, consider the function ' defined in Lemma 2.80. en show that 2 the inequalities < 0 < t imply that './ '.t/. Ä (ii) It follows from (i) that '.t/ '.1/ for all t .0; 1/. Similarly, '. 1/ '.t/ for all Ä 2 Ä t . 1; 0/. is easily yields the conclusion. 2 n Exercise 2.31. Fix x R , define P.x ˝/ w ˝ x w ˝ .x/ , and consider the N 2 NI WD f 2 j k N k D N g following function n fw .x/ x w for w ˝ and x R : WD k k 2 2 en we get from eorem 2.93 that @˝ .x/ co ˚@fw .x/ ˇ w P.x ˝/«: N D N ˇ 2 NI x w In the case where ˝ is not a singleton we have @fw .x/ N , and so N D ˝ .x/ N x co P.x ˝/ @˝ .x/ N NI : N D ˝ .x/ N e case where ˝ is a singleton is trivial. Exercise 2.33. (i) Fix any v Rn with v 1 and any w ˝. en 2 k k Ä 2 x; v ˝ .v/ x; v w; v x w; v x w v x w : h i Ä h i h i D h i Ä k k k k Ä k k It follows therefore that sup ˚ x; v ˝ .v/« d.x ˝/: v 1 h i Ä I k kÄ To verify the opposite inequality, observe first that it holds trivially if x ˝. In the case where 2 x ˝, let w ˘.x ˝/ and u x w; thus u d.x ˝/. Following the proof of Propo- … WD I WD k k D I sition 2.1, we obtain the estimate u; z u; x u 2 for all z ˝: h i Ä h i k k 2 Divide both sides by u and let v u= u . en ˝ .v/ v; x d.x ˝/, and so k k WD k k Ä h i I d.x ˝/ sup ˚ x; v ˝ .v/«: I Ä v 1 h i k kÄ EXERCISES FOR CHAPTER 3 Exercise 3.1. Suppose that f is Gâteaux differentiable at x with f .x/ v. en N G0 N D f .x td/ f .x/ f .x td/ f .x/ n fG0 .x/; v lim N C N lim N C N for all d R : h N i D t 0 t D t 0 t 2 ! ! C SOLUTIONS AND HINTS FOR EXERCISES 191 is implies by the definition of the directional derivative that f .x d/ v; d for all d Rn. 0 NI D h i 2 Conversely, suppose that f .x d/ v; d for all d Rn. en 0 NI D h i 2 f .x td/ f .x/ n f 0.x d/ v; d lim N C N for all d R : NI D h i D t 0 t 2 ! C n Moreover, f 0 .x d/ f 0.x d/ v; d v; d for all d R . us NI D NI D h i D h i 2 f .x td/ f .x/ n f 0 .x d/ v; d lim N C N for all d R : NI D h i D t 0 t 2 ! is implies that f is Gâteaux differentiable at x with f .x/ v. N G0 N D Exercise 3.2. (i) Since the function '.t/ t 2 is nondecreasing on Œ0; /, following the proof WD 1 of Corollary 2.62, we can show that if g Rn Œ0; / is a convex function, then @.g2/.x/ W ! 1 N D 2g.x/@g.x/ for every x Rn, where g2.x/ Œg.x/2. It follows from eorem 2.39 that N N N 2 WD ( 0 if x ˝; @f.x/ 2d.x ˝/@d.x ˝/ f g N 2 N D NI NI D ˚2Œx ˘.x ˝/« otherwise: N NI us in any case @f.x/ ˚2Œx ˘.x ˝/« is a singleton, which implies the differentiability of N D N NI f at x by eorem 3.3 and f .x/ 2Œx ˘.x ˝/. N r N D N NI (ii) Use eorem 2.39 in the out-of-set case and eorem 3.3. 1 2 Exercise 3.3. For every u F , the function fu.x/ Ax; u u is convex, and hence f 2 WD h i 2k k is a convex function by Proposition 1.43. It follows from the equality Ax; u x; A u that h i D h i fu.x/ Au: r D n 1 2 Fix x R and observe that the function gx.u/ Ax; u u has a unique maximizer on 2 WD h i 2k k F denoted by u.x/. By eorem 2.93 we have @f.x/ A u.x/ , and thus f is differentiable D f g at x by eorem 3.3. We refer the reader to [22] for more general results and their applications to optimization. n Exercise 3.5. Consider the set Y d R f 0.x d/ f 0 .x d/ and verify that Y is a WD f 2 j NI D NI g n subspace containing the standard orthogonal basis ei i 1; : : : ; n . us V R , which f j D g D implies the differentiability of f at x. Indeed, fix any v @f.x/ and get by eorem 2.84(iii) N 2 N that f .x d/ v; d for all d Rn. Hence @f.x/ is a singleton, which ensures that f is 0 NI D h i 2 N differentiable at x by eorem 3.3. N Exercise 3.6. Use Corollary 2.37 and follow the proof of eorem 2.40. 192 SOLUTIONS AND HINTS FOR EXERCISES Exercise 3.7. An appropriate example is given by the function f .x/ x4 on R. WD Exercise 3.10. Fix any sequence xk co ˝. By the Carathéodory theorem, find k;i 0 and f g wk;i ˝ for i 0; : : : ; n with 2 D n n xk X k;i wk;i ; X k;i 1 for every k N : D D 2 i 0 i 0 D D en use the boundedness of k;i k for i 0; : : : ; n and the compactness of ˝ to extract a f g D subsequence of xk converging to some element of co ˝. f g Exercise 3.16. By Corollary 2.18 we have N.x ˝1 ˝2/ N.x ˝1/ N.x ˝2/. Applying NI \ D NI C NI eorem 3.16 tells us that T.x ˝1 ˝2/ N.x ˝1 ˝2/ı NI \ D NI \ N.x ˝1/ N.x ˝2/ı N.x ˝1/ı N.x ˝2/ı D NI C NI D NI \ NI T.x ˝1/ T.x ˝2/: D NI \ NI Exercise 3.18. Consider the set ˝ .x; y/ R2 y x2 . WD f 2 j g Exercise 3.19. Fix any a; b Rn with a b. It follows from eorem 3.20 that there exists 2 ¤ c .a; b/ such that 2 f .b/ f .a/ v; b a for some c @f.c/: D h i 2 en f .b/ f .a/ v b a ` b a . j j Ä k k k k Ä k k Exercise 3.21. (i) F 0 R . (ii) F F . (iii) F R R . 1 D f g C 1 D 1 D C C (iv) F .x; y/ y x . 1 D f j j jg Exercise 3.23. (i) Apply the definition. (ii) A condition is A . B / 0 . 1 \ 1 D f g F Exercise 3.24. Suppose that T .x/ 0. en there exists a sequence of tk 0, tk 0, with ˝ D ! .x tkF/ ˝ for every k N. Choose fk F and wk ˝ such that x tkfk wk. C \ ¤ ; 2 2 2 C D Since F is bounded, the sequence wk converges to x, and so x ˝ since ˝ is closed. e f g 2 converse implication is trivial. Exercise 3.25. Follow the proof of eorem 3.37 and eorem 3.38. EXERCISES FOR CHAPTER 4 c Exercise 4.2. e set U L is closed by Proposition 4.4. D SOLUTIONS AND HINTS FOR EXERCISES 193 Exercise 4.3. Consider the l.s.c. function f .x/ 0 for x 0 and f .0/ 1. en the product WD ¤ WD of f f is not l.s.c. m Exercise 4.7. Solve the equation g.x/ 0, where g.x/ Pi 1 2.x ai /. r D r D D Exercise 4.8. Let ˝ x Rn Ax b . en N.x ˝/ AT Rp for x ˝. Corol- WD f 2 j D g I D f j 2 g 2 lary 4.15 tells us that x ˝ is an optimal solution to this problem if and only if there exists Rp 2 2 such that 2x AT and Ax b: D D Solving this system of equations yields x AT .AAT / 1b. D Exercise 4.11. e MATLAB function sign x gives us a subgradient of the function n f .x/ x 1 for all x R. Considering the set ˝ x R Ax b«, we have WD k k 2 WD f 2 j D ˘.x ˝/ x AT .AAT / 1.b Ax/, which can be found by minimizing the quadratic I D C function x u 2 over u ˝; see Exercise 4.8. k k 2 Exercise 4.12. If 0 @f.xk / for some k0 N, then ˛k 0, and hence xk xk S and 2 0 2 0 D D 0 2 f .xk/ V for all k k0. us we assume without loss of generality that 0 @f.xk/ as k N. D … 2 It follows from the proof of Proposition 4.23 that k k 2 2 2 2 0 xk 1 x x1 x 2 X ˛i Œf .xi / V X ˛i vi Ä k C Nk Ä k Nk C k k i 1 i 1 D D for any x S. Substituting here the formula for ˛k gives us N 2 k 2 k 2 2 2 Œf .xi / V Œf .xi / V 0 xk 1 x x1 x 2 X 2 X 2 Ä k C Nk Ä k Nk vi C vi i 1 i 1 D k k D k k k Œf .x / V 2 2 X i x1 x 2 and D k Nk vi i 1 D k k k 2 Œf .xi / V 2 X 2 x1 x : vi Ä k Nk i 1 D k k Since vi ` for every i, we get k k Ä k 2 Œf .xi / V 2 X x1 x ; `2 Ä k Nk i 1 D which implies in turn that k 2 2 2 XŒf .xi / V ` x1 x : (4.66) Ä k Nk i 1 D 194 SOLUTIONS AND HINTS FOR EXERCISES 2 en the series Pk1 1Œf .xk/ V is convergent. Hence f .xk/ V as k , which justifies D ! ! 1 (i). From (4.66) and the fact that Vk f .xj / for i 1; : : : ; k it follows that Ä 2 f g 2 2 2 k.Vk V/ ` x1 x : Ä k Nk ` x1 x us 0 Vk V k Nk. Since x was taken arbitrarily in S, we arrive at (ii). Ä Ä pk N Exercise 4.17. Define the MATLAB function m ai Pi 1 D x ai f .x/ k k ; x a1; : : : ; am WD m 1 … f g Pi 1 D x ai k k and f .x/ x for x a1; : : : ; am . Use the FOR loop with a stopping criteria to find WD 2 f g xk 1 f .xk/, where x0 is a starting point. C WD Exercise 4.18. Use the subdifferential Fermat rule. Consider the cases where m is odd and where m is even. In the first case the problem has a unique optimal solution. In the second case the problem has infinitely many solutions. Exercise 4.22. For the interval ˝ Œa; b R, the subdifferential formula for the distance func- WD tion 2.39 gives us 8 0 if x .a; b/; ˆf g N 2 ˆ ˆŒ0; 1 if x b; ˆ N D @d.x ˝/ < 1 if x > b; NI D f g N ˆŒ 1; 0 if x a; ˆ N D ˆ 1 if x < b: :ˆf g N Exercise 4.23. For ˝ IB.c r/ Rn, a subgradient of the function d.x ˝/ at x is WD I I N 80 if x ˝; v.x/ < x c N 2 N WD N if x ˝: : x c N … k N k en a subgradient of the function f can be found by the subdifferential sum rule of Corol- lary 2.46. Note that the qualification condition (2.29) holds automatically in this case. Exercise 4.27. Use the subgradient formula in the hint of Exercise 4.23 together with Propo- sition 2.54. Note that we can easily find an element i I.x/, and then any subgradient of the 2 N function fi .x/ d.x ˝i / at x belongs to @f.x/. WD I N N 195 Bibliography [1] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator eory in Hilbert Spaces, Springer, 2011. DOI: 10.1007/978-1-4419-9467-7. [2] D. P. Bertsekas, A. Nedić and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003. [3] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. DOI: 10.1017/CBO9780511804441. xi [4] J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization, Springer, 2000. DOI: 10.1007/978-1-4757-9859-3. xi [5] J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, Springer, 2005. [6] Z. Drezner and H. W. Hamacher (Eds.), Facility Location: Applications and eory, Springer, 2004. 154 [7] G. Colombo and P. R. Wolenski, “Variational analysis for a class of minimal time functions in Hilbert spaces,” J. Convex Anal., vol. 11, pp. 335–361, 2004. [8] J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Vols. I and II, Springer, 1993. DOI: 10.1007/978-3-662-06409-2. xi [9] J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer, 2001. DOI: 10.1007/978-3-642-56468-0. xi, 185 [10] H. W. Kuhn, “A note on Fermat-Torricelli problem,” Math. Program., vol. 54, pp. 98–107, 1973. DOI: 10.1007/BF01584648. 151 [11] Y. S. Kupitz, H. Martini and M. Spirova, “e Fermat-Torricelli problem, Part I: A dis- crete gradient-method approach,” Optim. eory Appl., vol. 158, pp. 305–327, 2013. DOI: 10.1007/s10957-013-0266-z. 154 [12] S. R. Lay, Convex Sets and eir Applications, Wiley & Sons, 1982. [13] G. G. Magaril-Il’yaev and V. M. Tikhomirov, Convex Analysis: eory and Applications, American Mathematical Society, 2003. 196 BIBLIOGRAPHY [14] B. S. Mordukhovich, Variational Analayis and Generalized Differentiation, Vols. I and II, Springer, 2006. DOI: 10.1007/3-540-31246-3. 40, 41, 49 [15] B. S. Mordukhovich and N. M. Nam, “Applications of variational analysis to a general- ized Fermat-Torricelli problem,” Optim. eory Appl., vol. 148, pp. 431–454, 2011. DOI: 10.1007/s10957-010-9761-7. 154 [16] B. S. Mordukhovich, N. M. Nam and J. Salinas, “Solving a generalized Heron problem by means of convex analysis,” Amer. Math. Monthly, vol. 119, pp. 87–99, 2012. DOI: 10.4169/amer.math.monthly.119.02.087. 154 [17] B. S. Mordukhovich, N. M. Nam and J. Salinas, “Applications of variational analysis to a generalized Heron problem,” Applic. Anal., vol. 91, pp. 1915–1942, 2012. DOI: 10.1080/00036811.2011.604849. 154 [18] B. S. Mordukhovich, N. M. Nam and C. Villalobos, “e smallest enclosing ball problem and the smallest intersecting ball problem: Existence and uniqueness of solutions,” Optim. Letters, vol. 154, pp. 768–791, 2012. DOI: 10.1007/s11590-012-0483-7. 168 [19] N. M. Nam and N. Hoang, “A generalized Sylvester problem and a generalized Fermat- Torricelli problem,” J. Convex Anal., vol. 20, pp. 669–687, 2013. 168 [20] N. M. Nam, N. Hoang and N. T. An, “Constructions of solutions to generalized Sylvester and Fermat-Torricelli problems for Euclidean balls,” Optim. eory Appl., in press. DOI: 10.1007/s10957-013-0366-9. 168 [21] Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer Academic Publishers, 2004. DOI: 10.1007/978-1-4419-8853-9. xi [22] Y. Nesterov, “Smooth minimization of nonsmooth functions,” Math Program., vol. 103, pp. 127–152, 2005. DOI: 10.1007/s10107-004-0552-5. 191 [23] J. F. Nash, Noncooperative Games, Doctoral thesis, Princeton University, 1950. 122 [24] J. F. Nash, “Equilibrium points in N-person games,” Proc. Nat. Acad. Sci. vol. 36, pp. 48–49, 1950. DOI: 10.1073/pnas.36.1.48. 122 [25] B. T. Polyak, An Introduction to Optimization, Optimization Solftware, 1987. [26] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. xi, 54 [27] R. T.Rockafellar and R. J-B. Wets, Variational Analysis, Springer, 1998. DOI: 10.1007/978- 3-642-02431-3. 49 BIBLIOGRAPHY 197 [28] R. T. Rockafellar, “Applications of convex variational analysis to Nash equilibrium,” Pro- ceedings of 7th International Conference on Nonlinear Analysis and Convex Analysis (Busan, Korea), pp. 173–183, 2011. 123 [29] A. Ruszczynski, Nonlinear Optimization, Princeton University Press, 2006. 154 [30] N. Z. Shor, Minimization Methods for Nondifferentiable Functions, Springer, 1985. DOI: 10.1007/978-3-642-82118-9. [31] J. J. Sylvester, “A question in the geometry of situation,” Quart. J. Math., vol. 1, p. 79, 1857. 168 [32] E. Weiszfeld, “Sur le point pour lequel la somme des distances de n points donnés est min- imum,” Tohoku Mathematics Journal, vol. 43, pp. 355–386, 1937. 150, 151 O [33] E. Zeidler, Applied Functional Analysis. Applications to Mathematical Physics, Springer, 1997. 123 199 Authors’ Biographies BORIS MORDUKHOVICH Boris Mordukhovich is Distinguished University Professor of Mathematics at Wayne State Uni- versity holding also Chair Professorship of Mathematics and Statistics at the King Fahd Univer- sity of Petroleum and Minerals. He has more than 350 publications including several mono- graphs. Among his best known achievements are the introduction and development of powerful constructions of generalized differentiation and their applications to broad classes of problems in variational analysis, optimization, equilibrium, control, economics, engineering, and other fields. Mordukhovich is a SIAM Fellow, an AMS Fellow, and a recipient of many international awards and honors including Doctor Honoris Causa degrees from six universities over the world. He is named a Highly Cited Researcher in Mathematics by the Institute of Scientific Information (ISI). His research has been supported by continued grants from the National Science Foundations. Department of Mathematics, Wayne State University, Detroit, MI 48202 [email protected]. Homepage: math.wayne.edu/ boris NGUYEN MAU NAM Nguyen Mau Nam received his B.S. from Hue University, Vietnam, in 1998 and a Ph.D. from Wayne State University in 2007. He is currently Assistant Professor of Mathematics at Portland State University. He has around 30 papers on convex and variational analysis, optimization, and their applications published in or accepted by high-level mathematical journals. His research is supported by a collaboration grant from the Simons Foundation. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97201 [email protected]. Homepage: web.pdx.edu/ mnn3 201 Index affine combination, 22 descent property, 151 affine hull, 21 dimension of convex set, 23 affine mapping, 7 directional derivative, 82 affine set, 21 distance function, 28 affinely independent, 23 domain, 11 asymptotic cone, 108 domain of multifunction, 18 biconjugacy equality, 80 epigraph, 11, 131 Bolzano-Weierstrass theorem, 3 Euclidean projection, 146 boundary of set, 3 Euclidean projection to set, 29 Brouwer’s fixed point theorem, 120 extended-real-valued function, 4 Carathéodory theorem, 99 Farkas lemma, 105 closed ball, 2 Farkas’ lemma, 99 closure of set, 3 Fenchel conjugate, 77 coderivative, 70 Fenchel-Moreau theorem, 80 coercivity, 134 Fermat rule, 56 compact set, 3 Fermat-Torricelli problem, 146 complement of set, 2 Fréchet differentiability, 55 cone, 33, 97 constrained optimization, 135 Gateaux differentiability, 95 O convex combination, 5 generalized ball, 168 convex combinations, 99 generalized Fermat-Torricelli problem, 154 convex cone, 33 generalized Sylvester problem, 169 convex cone generated by a set, 97 global/absolute minimum, 4 convex conic hull, 97 graph of multifunction, 18 convex function, 11 convex hull, 8, 99 Helly’s theorem, 102 convex optimization, 135 horizon cone, 108, 117 convex set, 5 hypograph, 179 derivative, 55 image, 44 202 INDEX indicator function, 54 positively homogeneous functions, 34 infimal convolution, 20, 75 projected subgradient method, 145 infimum, 4 interior of set, 2 Radon’s theorem, 102 relative interior, 25 Jensen inequality, 12 set separation, 39, 40 Karush-Kuhn-Tucker condition, 138 set separation, strict, 37 set-valued mapping/multifunction, 18 Lagrange function, 137 simplex, 24 level set, 34, 113, 131 singular subdifferential, 49 Lipschitz continuity, 114 Slater constraint qualification, 138 Lipschitz continuous functions, 49 smallest enclosing circle problem, 168 local minimum, 4 span, 23 locally Lipschitz functions, 49 strictly convex, 147, 155 lower semicontinuous, 178 strictly convex functions, 170 lower semicontinuous function, 129 strictly convex sets, 154 strictly monotone mappings, 125 marginal function, 72 strongly convex, 124 maximum function, 14, 86 strongly monotone mappings, 125 mean value theorem, 17, 106 subadditive functions, 34 minimal time function, 111 subdifferential, 53, 117 Minkowski gauge, 111, 113, 156 subgradient, 53 mixed strategies, 120 subgradient algorithm, 141 monotone mappings, 125 subgradients, 53, 117 Moreau-Rockafellar theorem, 63 subspace, 22 support function, 117 Nash equilibrium, 120 support functions, 75 noncooperative games, 120 supremum, 4 normal cone, 42, 117 supremum function, 16 open ball, 2 tangent cone, 104 optimal value function, 70 two-person games, 120 optimal value/marginal function, 18 upper semicontinuous function, 87 parallel subspace, 22 polar of a set, 103 Weierstrass existence theorem, 5 positive semidefinite matrix, 12 Weiszfeld algorithm, 150, 181