Convex Ana lysis
General Vector Spaces
C Zalinescu
World Scientific Convex Analysis i n General Vector Spaces This page is intentionally left blank Convex Analysis
1 n General Vector Spaces
C Zalinescu Faculty of Mathematics University "Al. I. Cuza" lasi, Romania
B% World Scientific • New Jersey • London • Singapore •• Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Zalinescu, C, 1952- Convex analysis in general vector spaces / C. Zalinescu p. cm. Includes bibliographical references and index. ISBN 9812380671 (alk. paper) 1. Convex functions. 2. Convex sets. 3. Functional analysis. 4. Vector spaces. I. Title.
QA331.5.Z34 2002 2002069000 515'.8-dc21
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Printed in Singapore by Uto-Print To the memory of my parents
Casandra and Vasile Zalinescu This page is intentionally left blank Preface
The text of this book has its origin in a course we delivered to students for Master Degree at the Faculty of Mathematics of the University "Al. I. Cuza" Ia§i, Romania. One can ask if another book on Convex Analysis is needed when there are many excellent books dedicated to this discipline like those written by R.T. Rockafellar (1970), J. Stoer and C. Witzgall (1970), J.-B. Hiriart- Urruty and C. Lemarechal (1993), J.M. Borwein and A. Lewis (2000) for finite dimensional spaces and by P.-J. Laurent (1972), I. Ekeland and R. Temam (1974), R.T. Rockafellar (1974), A.D. Ioffe and V.M. Tikhomirov (1974), V. Barbu and Th. Precupanu (1978, 1986), J.R. Giles (1982), R.R. Phelps (1989, 1993), D. Aze (1997) for infinite dimensional spaces. We think that such a book is necessary for taking into consideration new results concerning the validity of the formulas for conjugates and sub- differentials of convex functions constructed from other convex functions by operations which preserve convexity, results obtained in the last 10-15 years. Also, there are classes of convex functions like uniformly convex, uniformly smooth, well behaving, well conditioned functions that are not studied in other books. Characterizations of convex functions using other types of derivatives or subdifferentials than usual directional derivatives or Fenchel subdifferential are quite recent and deserve being included in a book. All these themes are treated in this book. We have chosen for studying convex functions the framework of locally convex spaces and the most general conditions met in the literature; even when restricted to normed vector spaces many results are stated in more general conditions than the corresponding ones in other books. To make
vii Vlll Preface this possible, in the first chapter we introduce several interiority and closed- ness conditions and state two strong open mapping theorems. In the second chapter, besides the usual characterizations and properties of convex functions we study new classes of such functions: cs-convex, cs- closed, cs-complete, lcs-closed, ideally convex, bcs-complete and li-convex functions, respectively; note that the classes of li-convex and lcs-closed functions have very good stability properties. This will give the possibility to have a rich calculus for the conjugate and the subdifferential of convex functions under mild conditions. In obtaining these results we use the method of perturbation functions introduced by R.T. Rockafellar. The main tool is the fundamental duality formula which is stated under very general conditions by using open mapping theorems. The framework of the third chapter is that of infinite dimensional normed vector spaces. Besides some classical results in convex analysis we give characterizations of convex functions using abstract subdifferentials and study differentiability of convex functions. Also, we introduce and study well-conditioned convex functions, uniformly convex and uniformly smooth convex functions and their applications to the study of the geometry of Banach spaces. In connection with well-conditioned functions we study the sets of weak sharp minima, well-behaved convex functions and global error bounds for convex inequality systems. The chapter ends with the study of monotone operators by using convex functions. Every chapter ends with exercises and bibliographical notes; there are more than 80 exercises. The statements of the exercises are generally ex tracted from auxiliary results in recent articles, but some of them are known results that deserve being included in a textbook, but which do not fit very well our aims. The complete solutions of all exercises are given. The book ends with an index of terms and a list of symbols and notations. Even if all the results with the exception of those in the first section are given with their complete proofs, for a successful reading of the book a good knowledge of topology and topological vector spaces is recommended. Finally I would like to thank Prof. J.-P. Penot and Prof. A. Gopfert for reading the manuscript, for their remarks and encouragements.
C. Zalinescu March 1, 2002 Ia§i, Romania Contents
Preface vii Introduction xi
Chapter 1 Preliminary Results on Functional Analysis 1 1.1 Preliminary notions and results 1 1.2 Closedness and interiority notions 9 1.3 Open mapping theorems 19 1.4 Variational principles 29 1.5 Exercises 34 1.6 Bibliographical notes 36
Chapter 2 Convex Analysis in Locally Convex Spaces 39 2.1 Convex functions 39 2.2 Semi-continuity of convex functions 60 2.3 Conjugate functions 75 2.4 The subdifferential of a convex function 79 2.5 The general problem of convex programming 99 2.6 Perturbed problems 106 2.7 The fundamental duality formula 113 2.8 Formulas for conjugates and e-subdifferentials, duality relations and optimality conditions 121 2.9 Convex optimization with constraints 136 2.10 A minimax theorem 143 2.11 Exercises 146 2.12 Bibliographical notes 155
ix x Contents
Chapter 3 Some Results and Applications of Convex Analy sis in Normed Spaces 159 3.1 Further fundamental results in convex analysis 159 3.2 Convexity and monotonicity of subdifferentials 169 3.3 Some classes of functions of a real variable and differentiability of convex functions 188 3.4 Well conditioned functions 195 3.5 Uniformly convex and uniformly smooth convex functions . . . 203 3.6 Uniformly convex and uniformly smooth convex functions on bounded sets 221 3.7 Applications to the geometry of normed spaces 226 3.8 Applications to the best approximation problem 237 3.9 Characterizations of convexity in terms of smoothness 243 3.10 Weak sharp minima, well-behaved functions and global error bounds for convex inequalities 248 3.11 Monotone multifunctions 269 3.12 Exercises 288 3.13 Bibliographical notes 292 Exercises - Solutions 297 Bibliography 349 Index 359 Symbols and Notations 363 Introduction
The primary aim of this book is to present the conjugate and subdifferential calculus using the method of perturbation functions in order to obtain the most general results in this field. The secondary aim is to give important applications of this calculus and of the properties of convex functions. Such applications are: the study of well-conditioned convex functions, uniformly convex and uniformly smooth convex functions, best approximation prob lems, characterizations of convexity, the study of the sets of weak sharp minima, well-behaved functions and the existence of global error bounds for convex inequalities, as well as the study of monotone multifunctions by using convex functions. The method of perturbation functions is based on the "fundamental duality theorem" which says that under certain conditions one has
inf $(i, 0) = max_ ( - $*(0,y*)). (FDF)
For many problems in convex optimization one can associate a useful perturbation function. We give here four examples; see [Rockafellar (1974)] for other interesting ones.
Example 1 (Convex programming; see Section 2.9) Let f,9i,---,9n '• X ->• E be proper convex functions with dom/ n C\7=i domgi ^ 0. The problem of minimizing f(x) over the set of those x S X satisfying gi{x) < 0 for alii = 1,..., n is equivalent to the minimization of $(x, 0) for x 6 X, where
$:IxF4l, *{x,y):={ f{x) * *(x)
(Y, ||-||) be a normed linear space, A : X —> Y be a linear operator, y0 £ Y and / : X —> M. be a proper convex function. Then f(x) + \\Ax + yo\\ > 0 for all x € X if and only if there exists y* € Y* such that f(x) — 2 (Tz + ?/o,2/*) - ||j/*||2 > 0 for all\ £ X. It is worth mentioning that adequate perturbation functions can be used for deriving formulas for the conjugate and e-subdifferential for many types of convex functions; this method is used by Rockafellar (1974) for foA and /l + • • • + fn, but we use it for almost all the operations which preserve convexity (see Section 2.8).
The formula (FDF) is automatically valid when infx€x $(a;,0) = -co and is equivalent to the subdifferentiability at 0 £ Y of the marginal func tion h : Y -> M, h(y) := mfxeX$(x,y), when infxex $(#,0) £ E. A sufficient condition for this is the continuity of the restriction of h to the affine hull of its domain at 0; note that 0 is in the relative algebraic interior of the domain of h in this case (without this condition one can give simple examples in which the subdifferential of h at 0 is empty). Considering the multifunction R:Ixl=j7 whose graph is the set grTl = {(x,t,y) | (x,y,t) £ epi$}, the continuity of /^(dom^) at 0 is ensured if It is relatively open at some (a;0,io) with (a;o,io,0) € gr7?.. This fact was observed for the first time by Robinson (1976). This remark shows the importance of open mapping theorems for convex multifunctions in con vex analysis. In Banach spaces such a result is the well-known Robinson- Introduction xm
Ursescu theorem. The preceding examples show that the consideration of more general spaces is natural: In Example 3 Y is a locally convex space while in Example 4 X can be endowed with the topology a(X, X'). The original result of Ursescu (1975) is stated in very general topological vec tor spaces. The inconvenient of Ursescu's theorem is that one asks the multifunction to be closed, condition which is quite strong in certain situ ations. For example, when calculating the conjugate or subdifferential of max(/, g) with /, g proper lower semicontinuous convex functions one has to evaluate conjugate or the subdifferential of 0 • / +1 • g which is not lower semicontinuous convex. Fortunately we dispose of another open mapping theorem in which the closedness condition is replaced by a weaker one, but one must pay for this by asking (slightly) more on the spaces involved. As said above, the second aim of the book is to give some interesting applications of conjugate and subdifferential calculus, less treated in other books. In many algorithms for the minimization problem (P) min f(x), s.t. x G
X, one obtains a sequence (xn) which is minimizing (i.e. (f(xn)) -» inf /) or stationary (i.e. (da/(z„)(0)) —> 0). It is important to know if such a sequence converges to a solution of (P). Assuming that S := argmin/ :=
{x | f(x) = inf/} 7^ 0, one says that / is well-conditioned if (ds(xn)) -» 0 whenever (xn) is a minimizing sequence, and / is well-behaved (asymptot ically) if (xn) is minimizing whenever (xn) is a stationary sequence; when 5 is a singleton well-conditioning reduces to the well-known notion of well- posedness in the sense of Tikhonov. If / is well-conditioned with linear rate the set argmin / is a set of weak sharp minima. When / is convex, we es tablish several characterizations of well-conditioning using the conjugate or the subdifferential of /. When 5 is a singleton one of the characterizations is close to uniform convexity of / at a point. One says that the proper function / : (X, ||-||) ->• K is strongly convex if
f(\x + (1 - X)y) < Xf(x) + (1 - X)f(y) - f A(l - A) ||z - yf for some c > 0 and for all x, y € dom/, A G [0,1]. This notion is not very adequate for non-Hilbert spaces; for general normed spaces, one says that / is uniformly convex if there exists p : 1+ -+ E+ with p(0) = 0 such that
f(Xx + (1 - X)y) < Xf(x) + (1 - X)f(y) - A(l - X)p (\\x - y\\) for all x,y e dom/ and all A € [0,1]. From a numerical point of view XIV Introduction
the class of uniformly convex functions is important because on a Banach space every uniformly convex and lower semicontinuous proper function has a unique minimum point and the corresponding minimization prob lem is well-conditioned. It turns out that uniformly convex functions have very nice characterizations using their conjugates and subdifferentials. The dual notion for uniformly convex function is that of uniformly smooth con vex function. An important fact is that / is uniformly convex (uniformly smooth) if and only if /* is uniformly smooth (uniformly convex). Another interesting application of convex analysis is in the study of monotone operators. This became possible by using a convex function associated to a multifunction introduced by M. Coodey and S. Simons. So, one obtains quite easily characterizations of maximal monotone operators, local boundedness of monotone operators and maximal monotonicity of the sum of two maximal monotone operators using continuity properties of convex functions, the formula for the subdifFerential of a sum of convex functions and a minimax theorem (whose proof is also included). A more detailed presentation of the book follows. The book is divided into three chapters, every chapter ending with ex ercises and bibliographical notes; there are more than 80 exercises. It also includes the complete solutions of the exercises, the bibliography, the list of notations and the index of terms. No prior knowledge of convex analysis is assumed, but basic knowledge of topology, linear spaces, topological (locally convex) linear spaces and normed spaces is needed. In Chapter 1, as a preliminary, we introduce the notions and results of functional analysis we need in the rest of the book. For easy reference, in Section 1.1 we recall several notions, notations and results (without proofs) which can be found in almost all books on functional analysis; let us men tion four separation theorems for convex sets, the Dieudonne and Alaoglu- Bourbaki theorems, as well as the bipolar theorem. In Section 1.2 we introduce cs-closed, cs-complete, lcs-closed (i.e. lower cs-closed) and ideally convex, bcs-complete, li-convex (i.e. lower ideally con vex) sets and prove several results concerning them. We point out the good stability properties of li-convex and lcs-closed sets. We also introduce two conditions denoted (Hx) and (Hwa;) which refer to sets in product spaces that are stronger than the cs-closedness and ideal convexity, but weaker than the cs-completeness and bcs-completeness of the sets, respectively. Then, besides the classical algebraic interior A1 and relative algebraic in- Introduction xv terior %A of a subset A of a linear space X, we introduce, when X is a topological vector space, the sets lcA and tbA, which reduce to lA when the affine hull aff A of A is closed or barreled, respectively, and are the empty set otherwise. The quasi interior of a set and united sets are also studied. In Section 1.3 we state and prove the famous Ursescu's theorem as well as a slight amelioration of Simons' open mapping theorem. As application of these results one reobtain the Banach-Steinhaus theorem and the closed graph theorem as well as two results of 0. Carja which are useful in control lability problems. Because the notions (with the exception of cs-closed and ideally convex sets) and results from Sections 1.2 and 1.3 are not treated in many books (to our knowledge only [Kusraev and Kutateladze (1995)] contains some similar material), we give complete proofs of the results. The chapter ends with Section 1.4 in which we state and prove the Ekeland's variational principle, the smooth variational principle of Borwein and Preiss, as well as two (dual) results of Ursescu which generalize Baire's theorem. Chapter 2 is dedicated, mainly, to conjugate and e-subdifferential calcu lus. Because no prior knowledge of convex analysis is assumed, we introduce in Section 2.1 convex functions, give several characterizations using the epi graph, or the gradients in case of differentiability, point out the operations which preserve convexity and study the important class of convex functions of one variable; the existence of the (e-)directional derivative and some of its properties are also studied. We close this section with a characterization of convex functions using the upper Dini directional derivative. Section 2.2 is dedicated to the study of continuity properties of convex functions. To the classes of sets introduced in Section 1.2 correspond cs- closed, cs-complete, lcs-closed, ideally convex, bcs-complete and li-convex functions. We mention the fact that almost all operations which preserve convexity also preserve the lcs-closedness and the li-convexity of functions as seen in Proposition 2.2.19. The most part of the results of this section are not present in other books; among them we mention the result on the convexity of a quasiconvex positively homogeneous function and the results on cs-closed, cs-complete, cs-convex, lcs-closed, ideally convex, bcs- complete and li-convex functions. Section 2.3 concerns conjugate functions; all the results are classical. Section 2.4 is dedicated to the introduction and study of direct proper ties of the subdifferential. Using such properties one obtains easily the for- xvi Introduction mulas for the subdifferentials of Af and /i • • • • •/„ which are valid without additional hypothesis. The classical theorem which states that the £-sub- differential of a proper convex function is nonempty and w*-compact at a continuity point of its domain, as well as the formula for the e-directional derivative as the support function of the e-subdifferential is also estab lished. The less classical result which states that the same formula holds for e > 0 when the function is not necessarily continuous (but is lower semi- continuous) is established, too. We mention also Theorem 2.4.14 related to the subdifferential of sublinear functions; some of its statements are not very spread. Other interesting results are introduced for completeness or further use. In Section 2.5 we introduce the general problem of convex program ming and establish sufficient conditions for the existence and uniqueness of solutions, respectively. We mention especially Theorems 2.5.2 and 2.5.5; Theorem 2.5.2 ameliorates a result of Polyak (1966), which shows that the refiexivity of the space, needed in proving the existence of solutions, is al most necessary, while Theorem 2.5.5 shows that the coercivity condition is essential for the existence of solutions. Section 2.6 is dedicated to perturbed functions. One introduces primal and dual problems, the marginal function, and give some direct properties of them. Then one obtains the formula for the e-subdifferential of the marginal function using the (e + ^-subdifferentials (with 77 > 0) of the perturbed function. Applying this result one obtains formulas for the e- subdifferential of several types of convex functions. In the main result of Section 2.7 we provide nine (non-independent) sufficient conditions which ensure the validity of the fundamental duality formula (FDF). The most known of them is that (x0,0) € dom$ and $(xo, •) is continuous at 0 for some xo£l. For the proof of the sufficiency of some conditions one uses the open mapping theorems established in Section 1.3. A related result involves also a convex multifunction; this will be useful for obtaining the formulas for the conjugate and the e-sub differential of a function of the forms go A with A a densely defined and closed linear operator and of g o H with g being increasing and H convex. Section 2.8 is dedicated entirely to conjugate and e-subdifferential cal culus for convex functions. The considered functions ip have the form: ip(x) = F(x, A{x)) and tp — f + g o A with A a continuous linear oper ator,
In Section 3.3 we introduce the class A of functions tp : K+ —> R+ with y>(0) = 0 and several useful subclasses. To any ip € A we associate tp# € A defined by In Section 3.7 we study the function /v : X ->• E, f^x) = /„" tp(t) dt, where norm. So, one establishes characterizations of the strict convex ity, the smoothness and the reflexivity of X by the strict convexity, the Gateaux differentiability of fv and the surjectivity of dfv, respectively. Introduction xix One obtains also characterizations of (local) uniform convexity and (local) uniform smoothness of X with the help of the properties of fv. For example one obtains: X is uniformly convex •£> X* is uniformly smooth •& fv is uni formly convex on bounded sets -O (fv)* is uniformly smooth on bounded sets •& {ftp)* is Frechet differentiable and V(/v)* is uniformly continuous on bounded sets. Note that a part of the results of this section can be found in the book [Cioranescu (1990)], but the proofs are different; note also that some notions are introduced differently in Cioranescu's book. Another application of convex analysis is emphasized in Section 3.8; here we apply the results on the existence, the uniqueness and the charac terizations of optimal solutions of convex programs to the problem of the best approximation with elements of a convex subset of a normed space. In Section 3.9 it is shown that there exists a strong relationship between the well-posedness of the minimization problem min/(:r) s.t. x £ X, and the differentiability at 0 of the conjugate /* of /; when / is convex these properties are equivalent. Using this result we establish a very interesting characterization of Chebyshev sets in Hilbert spaces and show that the class of weakly closed Chebyshev sets coincides with the class of closed convex sets in Hilbert spaces. Section 3.10 deals with sets of weak sharp minima, well-behaved convex functions and the study of the existence of global error bounds for convex inequalities. These notions were studied separately for a time, but they are intimately related. As noted above, argmin / is a set of weak sharp minima for / exactly when / is well-conditioned with linear rate. But the inequality f{x) < 0 has a global error bound exactly when argmin[/]+ is a set of weak sharp minima for [/]+ := max(/, 0). We give several characterizations of the fact that argmin / is a set of weak sharp minima for /, one of them being the fact that up to a constant, the conjugate /* is sublinear on a neighborhood of the origin. Several numbers associated to a convex function are introduced which are related to the conditioning number from numerical analysis. Although the most part of the results from this section are stated in the literature in finite dimensional spaces, we present them in infinite dimensions. The last section of this book, Section 3.11, is dedicated to the study of monotone multifunctions on Banach spaces. We use in the presentation two recent articles of Simons. The proofs are quite technical and use the lower semicontinuous convex function XM associated to the multifunction M : XX Introduction X =} X*, the minimax theorem and a few results of convex analysis. One obtains: two characterizations of maximal monotone multifunctions; the fact that the condition 0 6 int(domTi — domT2) is equivalent to other three conditions involving dom Tj and dom XT{ , and is sufficient for the maximal monotonicity of T\ + T2; dom T and Im T are convex if X is reflexive and T is maximal monotone; domT is convex if int(domT) ^ 0 and T is maximal monotone; T is locally bounded at XQ € (co(domX'))1 if T is a monotone multifunction; Rockafellar's theorem on the local boundedness of maximal monotone multifunctions. The result stating that for a maximal monotone multifunction T on the Banach space X for which dom T is convex the local boundedness of T at x 6 dom T implies that x € int (dom T) seems to be new. When applied to the subdifferential of a proper lower semicontinuous convex function / on the Banach space X, this result gives (for example): / is continuous &• dom/ is open <& df is locally bounded at any x £ dom/. The exercises are intended to exemplify the topics treated in the book. Many exercises are auxiliary results spread in recent articles, although some of them are extracted from other books. Some exercises are important results which could be parts of textbooks, but which do not fit very well with the aim of the present book. Among them we mention Exercise 3.11 on the Moreau regularization. Chapter 1 Preliminary Results on Functional Analysis 1.1 Preliminary Notions and Results In this section we introduce several notions and results on separation of sets as well as some properties of topological vector spaces and locally convex spaces which are frequently used throughout the book, for easy reference. Let X be a real linear (vector) space. Throughout this work we shall use the following notation (x, y being elements of X): [x, y] := {(1 — A)x + Xy \ A e [0,1]}, [x,y[:= {(1 - X)x + \y | A € [0,1[}, ]x,y[:= {(1 - X)x + Xy | A 6]0,1[}, called closed, semi-closed and open segment, respectively. Note that [x,x] =]x,x[= {x}\ If 0 ^ A, B C X, the Minkowski sum of A and B is A + B := {a + b | a £ A,b £ B). Moreover, if x £ X, X £ R and 0 ^ T C R, then x + A := A + x := A + {x}, A • A = {70 | A € A, a £ A} and XA — {A} • A. We shall consider that A + 0 = 0 and A • 0 = 0 • A = 0. A nonempty set A C X is star-shaped at a (G A) if [a, x] C A for all 2: £ A; A is convex if [x, y] C A for all x,y £ A; Ais a, cone if 1+ • A C A (in particular 0 € A when A is a cone), R+ is the set of nonnegative reals; A is afflne if Ax + (1 - X)y 6 A for all x,y e A and A e R; A is balanced if Ax € ^4 for all x e A and A £ [-1,1]; A is symmetric if A = -A. Hence ^4 is balanced if and only if A is symmetric and star-shaped at 0. We consider that the empty set is convex and afflne. It is easy to prove that A is afflne o3a6l, 3XQ linear subspace of X : A = a + X0 <=>Va€^4(3a€^4) : A — a is a linear subspace. When A is affine and a £ A, the linear space XQ := A — a is called the l 2 Preliminary results on functional analysis linear space parallel to A; we consider that the dimension of A is dimXo- Note that if (Aj)j€/ is a family of affine, convex, balanced subsets or cones of X then f]ieIAi has the same property (Exercise!); we use the usual convention that HieO7^ = -^- Taking into account this remark, we can introduce the notions of affine, convex and conic hull of a set. So, the affine, convex and conic hull of the subset A of X are: aff A := [){V C X \ A C V, V affine}, co A := P){C C X | A C C, C convex}, cone A := f]{C CX\AcC, C cone}, respectively. Of course, the linear hull of the subset A of X is the linear subspace spanned by A: linA := P|{^o C X \ A C X0, X0 linear subspace of X}. It is easy to verify (Exercise!) that aff A = { V" XiXi n G N, (Ai)i (Xi)l coA= {Yl^-^i n S N, (Aj) C K+, (xt) C A, Yl"-^ = 1) ' cone A - {Ax | A > 0, x £ A} = 1+ • A, where N is the set of positive integers. Let us mention some properties of the affine and convex hulls. Consider Y another linear space, T : X -> Y a linear operator, A,B c X, C C Y nonempty sets. Then: (i) aff(AxC) = aff Ax aff C; (ii) aff T(A) = T(aff A) (iii) aft(A + B) = aff A + aff B; (iv) Vo € A : aff A = a + aff (A - A) (v) aff (A - A) = UA>oA(A - A) if A is convex; (vi) aff A = linA if 0 e A (vii) co(A x C) = co A x coC; (viii) co T(A) = T(coA); (ix) co(A + 5) = co A + coB; (x) co(cone A) = cone(co A) (Exercises!). Let M C X be a linear subspace, and let A C X be nonempty; the algebraic interior of A with respect to M is aintM^:= {a 6 X | Vrr € M, 36 > 0, VA € [0,6] : a + Xxe A}. It is clear that aint^f Ac A and that M C aff (A - A) when aintM i^D- Preliminary notions and results 3 We distinguish two important cases: (i) M = X; in this case we write A% instead of aintM A; A1 is called the algebraic interior of A, (ii) M = aS(A — A); in this case aintM A is denoted by M and is called the relative algebraic interior of A. Therefore a £ A1 ii and only if aff A = X and a G 14 (Exercise!). When the set A is convex we have (Exercise!) that: VaGi : \m(A-a) = cone(A-A), whence aff A = a + cone(A — A) for every a £ A (hence cone(A — A) is the linear subspace parallel to aff A), aei'^Viel, 3A>0 : a + Xx £ A& cone(A - a) = X, and a G MoVa; G A, 3 A > 0 : (1 + X)a - Xx G A <$ cone(yl — a) = cone(A — A) <=> cone(A — a) is a linear subspace O M n(C — a) is a linear subspace; (1.1) the dimension of the convex set A is dim A := dim(aff A) = dim (cone(A — A)). Some properties of the algebraic interior are listed below. Let 0 ^ A,B CX, island A G E\ {0}; then: (i) *(x + A) =x + iA; (ii) ^XA) = X • % (iii) A + Bi C (A + B)'; (iv) A + Bl = {A + BY if Bi = B; (v) iA + iB c 1{A + B); (vi) *(A. + B) = A + *B if A, B are convex, VI / 0 and *B T^ 0; (vii) M 7^ 0 if A is convex and dim A < 00; (viii) if A is convex then [a, x[ C VI for all a G M. and i£A In the sequel the results will be established for real topological vector spaces (tvs for short) or real locally convex spaces (lcs for short). When X is a tvs it is well-known that the class Nx of closed and balanced neigh borhoods of 0 G X is a base of neighborhoods of 0; when X is a lcs then c the class J^ x of the closed, convex and balanced neighborhoods of 0 G X is also a base of neighborhoods of 0. If X, Y are real linear spaces, we denote by L(X, Y) the real linear space of linear operators from X into Y. The space L(X,R) is denoted by X' and is called the algebraic dual of X; an element of X' is called a linear functional. When X, Y are topological vector spaces, we denote by &{X, Y) 4 Preliminary results on functional analysis the linear space of continuous linear operators from X into Y; the space £(X, E) is denoted by X* and is called the topological dual of X. Let now A be an absorbing subset of the linear space X, i.e. 0 £ A1; the Minkowski gauge of A is defined by pA : X -> M, ^(a:) := inf{A > 0 | a; £ \A). It is obvious that PA = P[o,i].4- Moreover, if A, B C X and C C Y are absorbing and star-shaped sets, where Y is another linear space, one has (Exercise!): {x 6 X | pA(a) < 1} C A C {x € X | PA(Z) < 1}, VxGX : pAns(a;) = max {PA (a;), pB(a;)}, Va; 6 X, 2/ € Y : PAxc(x,y) = max{pA(x),pc(y)}- Other useful properties of the Minkowski gauge are mentioned in the next result. Recall that p : X -» M is sublinear if p(0) = 0, p(x + y) < p(x) +p(y) [with the convention (+oo) + (—oo) = +oo] and p(Xx) = Xp(x) for all x, y £ X, X € P := ]0, oo[; p is a semi-norm if p is a finite, sublinear and even [i.e. p(—x) = p(x) for every x € X] function. Proposition 1.1.1 Let A be a convex and absorbing subset of the linear space X. (i) Then PA is finite, sublinear and A1 = {x E X \ PA{X) < 1}; furthermore, if A is symmetric then PA is a semi-norm, too. (ii) Assume, moreover, that X is a topological vector space and V is a neighborhood of Q £ X. Then pv is continuous and intV = {xeX\pv{x) < 1}, clV = {xeX \pv{x) < 1}. The following result will be useful, too. Theorem 1.1.2 Let C be a convex subset of the topological vector space X. Then (i) clC is convex; (ii) if a £ int C and x £ cl C, then [a, x[ C int C; (iii) intC is convex; (iv) if int C ^ 0 then cl(int C) = cl C and int(cl C) = int C; (v) if int C ^ 0 then Cl = int C. Preliminary notions and results 5 Using the Minkowski gauge one obtains the geometrical versions of the Hahn-Banach theorem, i.e. separation theorems. In the sequel we give several separation theorems for convex subsets of topological vector spaces or locally convex space. Theorem 1.1.3 (Eidelheit) Let A and B be two nonempty convex subsets of the topological vector space X. If int A ^ 0 and B flint A = 0 then there exist x* e X* \ {0} and a £ E such that VxeA,\/yeB : (x,x*) < a < (y,x*), (1.2) or equivalently, supa;*(yl) < inf x*(B). The separation condition (1.2) can be given in a different manner. Let x* e X* \ {0} and a£t Consider the sets H^a:={x£X\(x,x*) H^,a~{xeX\ (x,x*) Hx*,a := {x e X \ (x,x*) = a}; similarly one defines H^. and H>.a. All these sets are convex and non empty. The set Hx*^a is called a closed hyperplane, H<. a and H>* a are called open half-spaces, while H-. a and H~.a are called closed half- spaces. Hx- A c H-, a and B C H-, a; in this situation we say that Hx*tC[ separates A and B; the separation is proper when AUB <£. Hx*i When x0 6 A and ffx*,a separates A and {xo} we say that ifx*,a is a supporting hyperplane of A at xo; XQ is called a support point and x* is called a support functional. Therefore x* € X* \ {0} is a support functional for A if and only if x* attains its supremum on A. Generally, Hx* In the case of locally convex spaces one has the following result for the separation of two sets. Theorem 1.1.5 Let X be a locally convex space and A,B C X be two nonempty convex sets. If A is closed, B is compact and An B = 0, then there exist x* G X* \ {0} and ct\,a.2 G K such that Vx G A, Vy e B : (x,x*) < ax < a2 < (y,x*), or equivalently, sup x* (A) < inf x* (B). The two preceding results can be stated in a more general setting. Theorem 1.1.6 Let A and B be two nonempty convex subsets of the topological vector space X such that int(A — B) ^ 0. Then 0 £ int(A-£)<£• 3x* GX*\{0} : supa;*(yl) < inf x*{B). Theorem 1.1.7 Let A and B be two nonempty convex subsets of the locally convex space X. Then Q$d{A-B)&3x* eX* : supx*{A) < inf x*(B). The preceding theorem shows the usefulness of having criteria for the closedness of the difference (or sum) of two convex sets. In order to give such a criterion, let A be a nonempty convex subset of the topological vector space X. The recession cone of A is defined by vecA := {u G X | Vo G A : a + uGA}. It is easy to show that rec A is a convex cone and A + rec A = A. When A is a closed convex set we have that vecA = f]t>ot(A-a) (1.3) for every a 6 A. In this case it is obvious that rec A is a closed convex cone which is also denoted by Aoo. It is easy to see that when X is a finite dimensional separated topological vector space and A is a closed convex nonempty subset of X, A^ = {0} if, and only if, A is bounded. This is no longer true when dimX ~ oo. Example 1.1.1 Let X := £p with p £ [l,oo] and A := {(x„)„>i G £p | \xn\ < n Vn G N}. It is obvious that A is a closed convex set which is not bounded because nen G A for every n G N, but A^ = {0}. Indeed, if Preliminary notions and results 7 u = (un) G Aoo then tu G A for every t > 0; so \tun\ < n for every t > 0, whence u„ = 0. Hence u = 0. The following famous theorem was obtained in [Dieudonne (1966)]. Theorem 1.1.8 (Dieudonne) Let A, B be nonempty closed convex subsets of the locally convex space X. If A or B is locally compact and A^ n J3oo is a linear subspace, then A — B is closed. In convex analysis (as well as in functional analysis) one often uses the following sets associated to a nonempty subset A of the locally convex space X: A° {x* eX*\Vx€A (x,x*)>-l}, A+ {x* ex* \Vxe A (x,x*)>0}, A^ {x* ex* IVze A (x,x*) = 0}, called the polar, the dual cone and the orthogonal space of A, respec tively. One verifies easily that A° is a w*-closed convex set which contains 0, that A+ is a w*-closed convex cone, and, finally, that A1- is a u;*-closed linear subspace of X*, where w* = a(X*,X) is the weak* topology on X*. Similarly, for 0 ^ B C X* we define the polar, the dual cone and the orthogonal space; for example, the polar of B is B° := {x G X | Vx* G B : (x,x*) > -1}. It is obvious that B° is a closed convex set containing 0, B+ is a closed convex cone, and, finally, B1- is a closed linear subspace. One verifies easily that when A,BcX and A G P we have: (i) A° is convex and 0 G A°; (ii) A U {0} C (A°)° =: A°°; (hi) AcB => A° D B°; (iv) (AUB)° = A°nB°; (v) if 0 € AnB then (A + B)+ = (AUB)+ =A+n B+; (vi) (\A)° = {A°; (vii) A° = A+ if A is a cone, and A° = A+ = A1- if A is a linear subspace; (viii) (T(A))° = (T*)"1^0), if T G &(X,Y), where Y is another locally convex space. A very useful result is the bipolar's theorem. Let X be a topological vector space and A C X; the set coA := cl(coA) is called the closed convex hull of the set A; it is the smallest closed convex set containing A. Similarly, coneA :— cl(coneyl) is called the closed conic hull of A. Theorem 1.1.9 (bipolar) Let A be a nonempty subset of the locally con- 8 Preliminary results on functional analysis vex space X. Then A00 = co(A U {0}), A++ = cone(co A), A±JL = cl(lin A). It follows that for the nonempty subset A of the lcs X one has: (a) A°° = A o A is closed, convex and 0 € A; (b) A++ = A <£> A is a closed convex cone; (c) A-"-1- = A <=> A is a closed linear subspace. Another famous result is the following. Theorem 1.1.10 (Alaoglu-Bourbaki) Let X be a locally convex space and U C X be a neighborhood of the origin. Then U° is w*-compact. We finish this preliminary section with some notions and results con cerning completeness and metrizability of topological vector spaces. The subset A of the topological vector space X is complete (quasi- complete) if every (bounded) Cauchy net (xi)i^i C A is convergent to an element x £ A. Of course, any complete set is closed and any closed subset of a complete set is complete (Exercise!). Recall that the topological space (X, T) is first countable if every element of X has a (at most) countable base of neighborhoods. Note that a subset A of a first countable tvs X is complete if and only if every Cauchy sequence of A is convergent to an element of A; in particular, a first countable tvs is complete if and only if it is quasi-complete (Exercise!). We shall use several times the hypothesis that a certain topological vector space is first countable. The next result refers to the first countability of locally convex spaces. Proposition 1.1.11 Let (X,T) be a locally convex space. Then (i) (X, T) is first countable O 3 7 a (at most) countable family of semi- norms on X such that r = rg> O r is semi-metrizable, i.e. there exists a semi-metric d on X such that T = T^; the semi-metric d may be chosen to be invariant to translations (i.e. d(x + z,y + z) = d(x,y) for allx,y,z € X). (ii) (X, T) is separated and first countable if and only if T is metrizable, i.e. there exists a metric d on X such that r = T&. Note that when the topology r of a locally convex space X coincides with the topology T 1.2 Closedness and Interiority Notions Xn Consider X a real topological vector space. We say that the series Yln>i is convergent (resp. Cauchy) if the sequence (£n)n£N is convergent (resp. x r Cauchy), where Sn := Y^k=i k f° every n £ N; of course, any convergent series is Cauchy. Let A C X; by a convex series with elements of A we mean a series = of the form Ylm>i ^mXm with (Am) C IR+, {xm) C A and Y,m>i ^™ ^ if, furthermore, the sequence (xm) is bounded we speak about a b-convex series. We say that A is cs-closed if any convergent convex series with elements of A has its sum in A;* A is cs-complete if any Cauchy convex series with elements of A is convergent and its sum is in A. Similarly, the set A is called ideally convex if any convergent b-convex series with elements of A has its sum in A and A is bcs-complete if any Cauchy b-convex series with elements of A is convergent and its sum is in A. It is obvious that any cs-closed set is ideally convex, every ideally convex set is convex, every cs- complete set is cs-closed and every complete convex set is cs-complete; if X is complete, then A C X is cs-complete (bcs-complete) if and only if A is cs- closed (ideally convex). If Xo is a linear subspace of X and A is a cs-closed (ideally convex) subset of X, then Xo C\ A is a cs-closed (ideally convex) subset of Xo (endowed with the induced topology). Moreover, if A C X and B C Y are nonempty, then A x B is cs-closed (cs-complete, ideally convex, bcs-complete) if and only if A and B are cs-closed (cs-complete, ideally convex, bcs-complete). If X is first countable, a linear subspace XQ of X is cs-closed (cs-complete) if and only if it is closed (complete); moreover, if X is a locally convex space, X0 is closed if and only if Xo is ideally convex. Note also that T(A) is cs-closed (cs-complete, ideally convex, bcs-complete) if A C X is cs-closed (cs-complete, ideally convex, 'Because X may not be separated, in fact we ask that every limit of (Sn) is in A. 10 Preliminary results on functional analysis bcs-complete) and T : X -> Y is an isomorphism of topological vector spaces (Exercise!), Y being another tvs. We consider that the empty set is convex, ideally convex, bcs-complete, cs-complete and cs-closed. It is worth to point out that when X is a locally convex space, every b-convex series with elements of X is Cauchy (Exercise!). The class of cs-closed sets (and consequently that of ideally convex sets) is larger than the class of closed convex sets, as the next result shows. Proposition 1.2.1 Let A C X be a nonempty convex set. (i) // A is closed or open then A is cs-closed. (ii) If X is separated and dim A < oo then A is cs-closed. Proof, (i) Let £n >! Anin be a convergent convex series with elements of A; denote by x its sum. Suppose that A is closed and fix a £ A. Then, for every n G N we have that X)fc=i ^kXk + (l — Yl'kLn+i ^*) a S A. Taking the limit for n -> oo, we obtain that x G cl A = A. Suppose now that A is open. Assume that x $. A. By Theorem 1.1.3, there exists x* G X* such that (a - x, x") > 0 for every a G A. In particular (xn — x,x*) > 0 for every n G N. Multiplying by \n > 0 and adding for n G N we get (since A„ > 0 for some n) the contradiction XnXn x An x = 0 < ^n>1 A„ (xn - x, x*) = (X]„>i ' *)~ \52n>i ) ^' ") °' Therefore x G A. So in both cases A is cs-closed. (ii) We prove the statement by mathematical induction on n := dim A. If n = 0 A reduces to a point; it is obvious that A is cs-closed in this case. Suppose that the statement is true if dim A < n G N U {0} and show it for dim A = n + 1. Without any loss of generality we suppose that 0 G A; then Xo := aff A is a linear subspace with dimX0 = n + 1. Because on a finite dimensional linear space there exists a unique separated linear topology and in such spaces the interior and the algebraic interior i coincide for convex sets, we have that A — intx0 A ^ 0. Let J2n>i ^n%n be a convergent convex series with elements of A and sum x. Assume that x fi A. Because A is convex, the set P :— {n G N | A„ > 0} is infinite, and so we may assume that P — N. Applying now Theorem 1.1.3 in XQ, there exists XQ G XQ \ {0} such that (x — x, XQ) > 0 for every x G A. But X)n>i An (xn —X,XQ) = 0. Since {xn —X,XQ) > 0 and Xn > 0 for every n, we obtain that (xn) C AQ := A(~)Hx*t\, where A :— (X,XQ). Since Closedness and inferiority notions 11 dimAo < dimHx*t\ = n, from the induction hypothesis we obtain the contradiction x e A0 c A. Therefore x € A. The proof is complete. • Other properties of cs-closed and ideally convex sets are given in the following result. Proposition 1.2.2 (i) If Ai C X is cs-closed (resp. ideally convex) for every i £ I then P|ie/ A{ is cs-closed (resp. ideally convex). (ii) // Xi is a topological vector space and Ai C Xi is cs-closed (resp. ideally convex) for every i 6 I, then Ylizi -^-i is cs-closed (resp. ideally convex) in Yliei Xi (which is endowed with the product topology). Proof. The proof of (i) is immediate, while for (ii) one must take into account that a sequence (xn)n€N C X := YlieI Xi converges toxGX (resp. l l is bounded) if and only if (x n) converges to x in Xi (resp. is bounded) for every i £ I. • We say that the subset C of Y is lower cs-closed (Ics-closed for short) if there exist a Frechet space X and a cs-closed subset B of X x Y such that C = Pry (B). Similarly, the subset C of Y is lower ideally convex (li-convex for short) if there exist a Frechet space X and an ideally convex subset B of X xY such that C = Pry (B). It is obvious that any cs-closed (resp. ideally convex) set is Ics-closed (resp. li-convex), any Ics-closed set is li-convex and any li-convex set is convex, but the converse implications are not true, generally. Note also that T(A) is Ics-closed (resp. li-convex) if A C X is Ics-closed (resp. li-convex) and T : X -> Y is an isomorphism of topological vector spaces (Exercise!). The classes of Ics-closed and li-convex sets have very good stability properties as the following results show. We give only the proofs for the "li-convex" case, that for the "Ics-closed" case being similar. Proposition 1.2.3 Suppose that Y is a Frechet space and C CF x Z is a li-convex (Ics-closed) set. Then Prz(C) is a li-convex (Ics-closed) set. Proof. By hypothesis, there exists a Frechet space X and an ideally convex subset B C X x (Y x Z) such that C = Pryxz(B)- Since Ix7is a Frechet space and Prz(C) = Prz(B), we have that Prz(C) is a li-convex subset of Z. • Proposition 1.2.4 Let I be an at most countable nonempty set. (i) If Ci C Y is li-convex (Ics-closed) for every i £ I then (~)ieI Ci is li-convex (Ics-closed). 12 Preliminary results on functional analysis (ii) // Y{ is a topological vector space and Ci C Yi is li-convex (Ics- closed) for every i 6 I, then Yli&iCi 8S li-convex (Ics-closed) in W^jYi. Proof, (i) For each i £ I there exist X» a Frechet space and an ideally convex set Bt C X» xY such that d = PrY(Bi). The space X := fliei -%i is a Frechet space as the product of an at most countable family of Frechet spaces. Let Bi •= {{(xj)jei,y) e X x Y | (xi,y) £ Bi} . Then Bi is an ideally convex set by Proposition 1.2.2(h). It follows that B := f]ieI Bi is ideally convex by Proposition 1.2.2(i). Since PrY(B) = Hie/ Ci, ^ f°uows that flie/ Ci is li-convex. (ii) For each i £ I there exist Xj a Frechet space and an ideally convex set Bi cXiX Yi such that d = Pry; (Bi). By Proposition 1.2.2(h), ]JieI Bt is an ideally convex subset of riie/(^j x ^)- The space X := riig/ ^i 1S a Frechet space; let Y := FJie/ ^i- Consider the set B:= {((xi)ieI,(yi)ieI) £XxY\ (xi,yi) £ Bi Vt 6 /} . X Since T : Ilie/( * x y4) -• X x y, T((xi,yi)i6/) := ((xi)ieI,(yi)iei) is an isomorphism of topological vector spaces, B — T (Y\ieI Bi) is ideally = convex. As C := riig/ C* ^VY(B), C is li-convex. D Before stating other properties of li-convex and lcs-closed sets, let us define some notions and notations related to multifunctions. Let E,F be two nonempty sets; a mapping ft : E —• 2F is called a multifunction, and it will be denoted by ft : E =X F. The set domft := {x £ E \ 5l(x) -£ 0} is called the domain of the multifunction 51; the image of 51 is Imft := UigB^(a;)i the graph of 51 is the set grft :— {(x,y) \ y £ 5l(x)} C E x F; the inverse of the multifunction 51 is the multifunction ft"1 : F =} E defined by ft_1(?/) := {x £ E | y £ %(x)}. Therefore domft-1 = Imft, Imft-1 = domft and grft-1 = {(y,x) | (x,y) £ grft}. Frequently we shall identify a multifunction with its graph. For A C E and 1 -1 in B C F one defines 51(A) := \JxeA5l(x) and ft" ^) := Uj,eB# (2/); particular Imft = ft(.E) and domft = ft_1(F). If 8 : F =4 G is another multifunction, then the composition of the multifunctions S and 51 is the multifunction Soft :E=lG, (Soft) (a) := \Jy€Ci{x) S(y). If ft, S : E =} F and F is a linear space, the sum of ft and S is the multifunction ft + S : E =t F, (ft + S)(x) :=5i(x) + S(x). Closedness and interiority notions 13 Let now K : X r{ 7; we say that 51 is convex (closed, ideally con vex, bcs-complete, cs-convex, cs-complete, li-convex, lcs-closed) if its graph is a convex (closed, ideally convex, bcs-complete, cs-convex, cs-complete, li-convex, lcs-closed) subset of X x Y. Note that 31 is convex if and only if Vi,a' eX, VAe [0,1] : \3l(x) + (l-\)3l(x')c3l(\x + {l-\)x'). Proposition 1.2.5 Let A, B C X, 31, S : X =4 Y and7:Y =} Z. (i) If X is a Frechet space and A, 31 are li-convex (resp. lcs-closed), then 01(A) is li-convex (resp. lcs-closed). (ii) If X is a Frechet space and A, B are li-convex (resp. lcs-closed), then A + B is li-convex (resp. lcs-closed). (hi) If Y is a Frechet space and 31, T are li-convex (resp. lcs-closed), then T o 31 is li-convex (resp. lcs-closed). (iv) IfY is a Frechet space and 31,$ are li-convex (resp. lcs-closed), then 31 + § is li-convex (resp. lcs-closed). Proof, (i) We have that %{A) =PrY((AxY)ngrJl). Using successively Propositions 1.2.4(h), 1.2.4(i) and 1.2.3, it follows that 31(A) is li-convex. (ii) Let T : X xX —> X, T(x,y) := x + y. Since T is a continuous linear operator, grT is a closed linear subspace; in particular T is a li-convex multifunction. Since Ax B is li-convex, by (i) A + B = T(A x B) is a li-convex set. (hi) We have that gr(To3?) = PrXx4(gr:R x Z) n (X x grT)). The conclusion follows from Propositions 1.2.4(h), 1.2.4(i) and 1.2.3. (iv) The sets T~{(x,z,y,y')\x€X, y,y' eY, z = y + y'} C (X x Y) x (Y x Y), A:= {(x,z,y,y') \ (x,y) G gr#, y',z € Y} and B := {(x,z,y,y') \ (x,y') G grS, y,z € Y} are li-convex sets (the first being a closed linear subspace). Therefore gr(# + S) = PrXxY (THAHB) is li-convex. • 14 Preliminary results on functional analysis Let Y be another topological vector space and A C X xY; we introduce the conditions (Ha;) and (Hwa;) below, where x refers to the component xeX: (Ha:) If the sequences ((xn, yn)) C A and (A„)„>! C ffi+ are such that E„>i K = 1, En>i ^nVn^as sum y and X)„>i A«a;n is Cauchy, x s then the series X^n>i ^n n * convergent and its sum x G X verifies {x,y)EA. (Hwi) If the sequences ({xn,yn))n>1 C A and (An)n>i C K+ are such that ((xn,yn)) is bounded, En>x An = 1, £„>i A„2/n has sum y x s and ^ra>1 Anxn is Cauchy, then the series X^n>1 ^« « ^ convergent and its sum x € X verifies (x, y) 6 A. Of course, when X is a locally convex space, deleting "^2n>1 Ana;n is Cauchy" in condition (Hwa;) one obtains an equivalent statement. In the next result we mention the relationships among conditions (Ha;), (Hwi), ideal convexity, cs-closedness, cs-completeness and convexity. The proof being very easy we omit it. Proposition 1.2.6 Let A C X xY and BcXxYxZbe nonempty sets. (i) Assume that Y is complete. Then B satisfies (H.(x,y)) if and only if B satisfies (Ha;); B satisfies (Hwa;) if and only if B satisfies (Hw(x,j/)). (ii) Assume that X is complete. Then A satisfies (Ha;) if and only if A is cs-closed; A satisfies (Hwa;) if and only if A is ideally convex. (iii) Assume that Y is complete. Then A satisfies (Ha;) if and only if A is cs-complete; A satisfies (Hwa;) if and only if A is bcs-complete. (iv) If A satisfies (Ha;) then A satisfies (Hwx); if A satisfies (Hwa;) then A is convex. (v) Assume that X is a locally convex space and Prx(A) is bounded. If A satisfies (Ha;) then Pry(^4) is cs-closed; if A satisfies (Hwa;) then Pry (A) is ideally convex. We define now several interiority notions. Let 0 / A c X. We denote by rint A the interior of A with respect to aff A, i.e. rint A := intafj A A. Of Closedness and inferiority notions 15 course, rint A C %A. Consider also the sets l c . . A if aff A is a closed set, 1 otherwise, rint A if aff A is a closed set, riA:= | 0 otherwise, l i6 . | A if XQ is a barreled linear subspace, -{ otherwise, where X0 = \in(A — a) for some (every) a G A; XQ is the linear subspace, parallel to aff A. In the sequel, in this section, A C X is a nonempty convex set. Taking into account the characterization (1.1) of *A, we obtain that x € tcA <£> cone(A — x) is a closed linear subspace of X & M X(A — x) is a closed linear subspace of X, and x £ A -£> cone(^4 — a;) is a barreled linear subspace of X <£> M n(A — x) is a barreled linear subspace of X. If X is a Frechet space and aff A is closed then %CA = lbA, but it is possible to have ibA ^ 0 and icA = 0 (if aff A is not closed). The quasi relative interior of A is the set qri A := {x € A | cone(yl — z) is a linear subspace of X}. Taking into account that in a finite dimensional separated topological vector space the closure of a convex cone C is a linear subspace if and only if C is a linear subspace (Exercise!), it follows that in this case qri A = %A = ic A = ibA Below we collect several properties of the quasi relative interior. Proposition 1.2.7 Let A C X be a nonempty convex set and T £ L{X,Y). Then: a £ qri A •& a G A and cone(A — a) = cone(A - A) OaEi and a - A C cone(yi - a), (1.4) iA C qriA = AnqriA, VaGqriA, Vx £ A : [a,z[CqriA, (1.5) 16 Preliminary results on functional analysis and T(qriA) C qriT(A). In particular qriA is a convex set. Assume that qriA ^ 0; then qriA = A and *(T(A)) C T(qriA) C qriT(A) C T(A) c T(qriA). (1.6) Moreover, if Y is separated and finite dimensional then T(qiiA)=i(T(A)). (1.7) Proof. The first equivalence in Eq. (1.4) is immediate from the definition of qriA. Of course, cone (A — a) = cone (A — A) implies that a — A C cone(A - a). Conversely, if a — A C cone(A - a) then -cone(A - a) = cone(a — A) C cone (A — a), which shows that cone (A — a) is a linear subspace. Therefore Eq. (1.4) holds. If a £ lA, from Eq. (1.1) we have that a £ A and cone(A — a) is a linear subspace, and so cone(A — a) is a linear subspace, too. Hence a € qriA. The equality qriA = An qriA follows immediately from the relation coneC = cone C, valid for every nonempty subset C of X. Let a € qri A, x e A, A € [0,1[ and a\ :— (1 — X)a + Xx. Then A-ADA-aA = (l- A)(A - a) + A(A - z) D (1 - A)(A - a), and so, taking into account Eq. (1.4), we have cone(A - A) D cone(A - a\) D cone ((1 - A)(A - a)) = cone (A — a) = cone (A — A). Therefore cone(A — A) = cone(A — a^)- Since a\ € A, from Eq. (1.4) we obtain that a\ G qriA. The proof of Eq. (1.5) is complete. Let a £ qriA; then, by Eq. (1.4), a - A C cone(A — a), and so Ta - T{A) = T(a-A)cT (cone(A - a)) C T (cone(A - a)) = cone (T(A) - Ta), which shows that Ta € qriT(A). Assume now that qriA ^ 0 and fix a £ qriA. It is sufficient to show Eq. (1.6); then the equality qriA = A follows immediately from the last inclusion in Eq. (1.6) for T = Idx and from Eq. (1.5). Let y e i(T'(A)); using Eq. (1.1), there exists A > 0 such that (1 + X)y - XT a € T(A). So, (1 + X)y - XTa = Tx for some x £ A. It follows that _1 y = Txx, where xx ~ (1 + A) (Aa + x). But, using Eq. (1.4), x\ 6 qriA, and so y 6 T(qriA). Closedness and inferiority notions 17 The second inclusion in Eq. (1.6) was already proved, while the third is obvious. So, let y G T(A); there exists x G A such that y = Tx. By Eq. (1.5), (1-X)a + Xx G qriAfor A G]0,1[, and so (l-X)Ta + Xy G T(qriA) for A G ]0,1[. Taking the limit for A —> 1, we obtain that y G T(qri A). When Y is separated and finite dimensional we have (as already ob served) that i(T(A)) = qriT(A); then Eq. (1.7) follows immediately from Eq. (1.6). • The notion of quasi relative interior is related to that of united sets. Let X be a locally convex space and A,B C X he nonempty convex sets; we say that A and B are united if they cannot be properly separated, i.e. if every closed hyperplane which separates A and B contains both of them. The next result is related to the above notions. Proposition 1.2.8 Let X be a locally convex space, A,BcX be non empty convex sets and x G X. (i) A and B are united O- cone(A-B) is a linear subspace •& (A — B)~ is a linear subspace. (ii) Assume that cone (A — x) is a linear subspace. Then x G c\A. Moreover, i/aff A is closed and rint A ^ 0 then ~x G rint A. Proof, (i) Assume that A and B are united but C := cone(A — B) is not a linear subspace. Then there exists XQ G (—C) \ C. By Theorem 1.1.5 there exists x* G X* such that {x0, x*) < 0 < (z, x*) for every z G C. Then 0 < (x - y,x*) for all x G A, y G B, and so (x,x*) < X < (y,x*) for all x G A, y G B, for some A G M. Therefore Hx* t\ separates A and B. It follows that (x,x*) = X = (y,x*) for all x G A, y G B, and so 0 < (z,x*) for every z G C. Thus we have the contradiction (x0,x*) = 0. Therefore cone (A — B) is a linear subspace. Let C := cone (A — B) be a linear subspace and (x,x*) < X < (y, x") for all x G A, y G B, for some x* G X* and Aei. Then 0 < (z,x*) for every z G A — B, and so 0 < (z, x*) for z € C. Then, since C is a linear subspace, 0 = (z,x*) for every z G C which implies immediately that Hx*,\ contains A and B. Therefore A and B are united. The other equivalence is an immediate consequence of the bipolar the orem (Theorem 1.1.9). (ii) By (i) we have that {x} and A are united. Assuming that x $ cl A, using again Theorem 1.1.5, we obtain that {x} and A can be properly separated. This contradiction proves that x E. cl A. 18 Preliminary results on functional analysis Suppose now that aSA is closed and rint A / 0. Without loss of gen erality we suppose that x = 0. By what was shown above we have that x = 0 € clA C cl(affA) = aff A. Thus X0 := aS A is a linear space. Assuming now that 0 £ intx0 A ^ 0, we obtain that {x} and A can be properly separated (in XQ, and therefore in X) using Theorem 1.1.3. This contradiction proves that x € rint A. • From the preceding proposition we obtain that when X is a locally convex space and A C X is a nonempty convex set, the quasi relative interior of A is given by the formula qri A = A (~1 {x € X \ {x} and A are united}. The next result shows that the class of convex sets with nonempty quasi relative interior is large enough. Proposition 1.2.9 Let X be a first countable separable locally convex space and A C X be a nonempty cs-complete set. Then qri A ^ 0. Proof. Since X is first countable, by Proposition 1.1.11, the topology of X is determined by a countable family 7 = {pn \ n 6 N} of semi-norms. Without loss of generality we suppose that pn < pn+i for every n € N. Since Ty is semi-metrizable, the set A is separable, too. Let A0 = {xn | n € N} C n _n A be such that A C cl A0. Consider j3n € ]0,2~ ] such that /3npn(xn) < 2 . The series ^n>1 Pnxn is Cauchy (since for m > n and peNwe have that PniT^Z^Xk) < ET=mPkPn(xk) < ET=Z^Pk(xk) < 2"™+!). Taking 1 1 s a ^n '•= (Z)n>i Pn) Pn, ]Cn>i ^n " * Cauchy convex series with elements of A. Because A is cs-complete, the series ^n>1 Xnxn is convergent and its sum x € A. Suppose that x ^ qri A. Then there exists XQ € (—C)\C, where C := cb~m(A — x). Using Theorem 1.1.5, there exists x* € X* such that (xo,x*) < 0 < {z,x*) for all z € C. In particular ( *) > 0 for — = every x £ A. But En>i ^» (^ ^^o) 0- Since {xn — X,XQ) > 0 and A„ > 0 for every n, we obtain that (x - x, x*) = 0 for every x € AQ. Since AQ is dense in A and x* is continuous, we have that {x — x, x*) =0 for every x € A, and so (z,a;*) = 0 for every z E C. Thus we get the contradiction (x0,x*) = 0. Therefore qri A ^ 0. D Open mapping theorems 19 1.3 Open Mapping Theorems Throughout this section the spaces X, Y are topological vector spaces if not stated otherwise. We begin with some auxiliary results. Lemma 1.3.1 Let X,Y be first countable topological vector spaces, 31 : X =$ Y be a multifunction and XQ G X. Suppose that grIR satisfies condi tion (Hwa;). Then where Nxl^o) ** the class of all neighborhoods of XQ. int Proof. Let y0 G C\u£7fx(x0) (clft(t/)). Replacing grft by gv% - {x0,y0) if necessary, we may assume that (xo,yo) = (0,0). Let U G Nx- Since X is first countable, there exists a base (Un)n>i C Nx of neighbor hoods of 0 such that Un + Un C Un-i for every n > 1, where [7o := ?/• Because 0 G f")^^ int (cl#([/)), there exists (Vn)n>i C Ny such that V„ C int (cl3i([7n)) for every n > 1. Since Y is first countable, we may suppose that (Vn)n>i is a base of neighborhoods of 0 G Y and, moreover, yn+i + Vn+i C KJ for every n > 1. Consider j/' G int (clCR.(C/i)); there exists fi G]0,1[ such that y := (1 — lJ)~1y' £ cl3?([/i). There exists (£1,2/1) e gr$ such that x\ G C/i and -1 2/ - 2/i G M^2- It follows that /x (y - j/i) G V2 C cl!R(t/2)- There exists (^2,2/2) G grR such that xi G L/2 and /x_1(2/ _ 2/i) — 2/2 G M^3- It follows that /U~2y — fJ-~2yi — H~lyi 6 V3 C c\"R{Uz). Continuing in this way we m+x m+1 find ({xm,ym))m>1 C grR such that xm e Um and n~ y - n~ yi - -m+2 M 2/2 - ••• - ym G fiVm+i for every m > 1. Therefore um := y - t ,_1 m 2/1-/^2/2 A " 2/m G ^ Ki+i C Vm+i for every m > 1. Moreover, m_1 m x m M 2/m = vm-i -vm e \i ~ Vm - fi Vm+1, and so ym £ Vro - fiVm+1 C Vm + Vm C Vm_i for m > 2. It follows that (xm)m>i -> 0, (ym)m>i -> 0 m l and (um)m>i -> 0, whence J2m>i n ~ ym has sum y. Taking Am := m_1 A A has (l-/i)^ , we have that (Am)m>i C R)., Em>i ™ = *> Em>i m2/m sum (1 - ju)y = y', the sequence ((xTO,ym)) is bounded (being convergent) and, because £mtf„+i Ama;TO € C/„+i + Un+2 + ••• + Un+P C f7„+1 + Un+1 C f/n for every n > 1, the series 5Zm>1 Amxm is Cauchy. By hypothesis there exists x' £ X, sum of the series ]Cm>i Am#m> such that (x',y') £ grX Let l m lx x := (1 - n)~ x'- We have that YZi=i H ~ m G t/i + U2 + • • • + Un C t/i -I- Ui C t/. Since U is closed we have that x E.U, and so a;' G (1 — n)U C 20 Preliminary results on functional analysis U. Thus y' £ "R{U). Therefore int (cl3l(l7i)) C R(U). In particular 0 € int%(U). The proof is complete. • Note that we didn't use the fact that X or Y is separated. Observe also nt c that it is no need that x0 £ dom"R when y0 £ f\ue^x(xo) * ( ^(^0)' but, necessarily, xo £ cl(domD?) and yo £ int(ImlR) in our conditions. Note also that condition (Hwa;) may be weakened by asking that the sequence ((xm,ym)) C A be convergent instead of being bounded. In the case of normed spaces one has the following variant of the result in Lemma 1.3.1. Having the normed vector space (nvs) (X, ||-||), we denote by Ux the closed unit ball {x £ X \ \\x\\ < 1}, by Bx the open unit ball {x € X | ||a;|| < 1} and by Sx the unit sphere {x 6 X | ||a;|| = 1}. Lemma 1.3.2 Let (X, || • ||), (Y, || • ||) be two normed linear spaces and let 31: X =4 Y be a multifunction. Suppose that condition (Hwa;) holds and (x0,y0) &X xY. If yo + nUy C cl (^(XQ + pUx)), where rj,p > 0, then yo + nBy C%(x0 + pUx). Proof. We may take (a;o,2/o) = (0,0). One follows the same argument as in the proof of the preceding lemma, but with Un := pUx and Vn := nBy for n > 1. Consider y' 6 nBy and take p, £]0,1[ such that y :— (1 — p) y' £ c\"R(pUx)- We find the sequence {{xn,yn))n>l C grft such that _1 n (xn) C pUx and v„ := y - 2/1 - py2 ^" 2/n £ p r)BY for n > 1. _1 Hence (un) -> 0 and /x" 2/„ = u„_i - vn, whence \\yn\\ < 77(1 + p) for n 1 = ne ser es n > 1. Taking An := (1 — p)p ^ > 0 for n > 1, X)n>i ^" 1> * i S«>i ^n%n is Cauchy and the series X3n>i ^n2/n is convergent with sum y'. Since 51 satisfies the condition (Hwi), we obtain that the series J2n>i ^nXn is convergent with sum x' and (x',y') £ gr!R. Of course, x' £ pUx- Hence nBy C npUx)- • Lemma 1.3.3 Let X,Y be topological vector spaces, 3? : X =4 Y be a closed convex multifunction and XQ € X. Suppose that X is complete and first countable. Then condition (1.8) holds. Open mapping theorems 21 int cl U Proof. Let y0 G f)u£Nx(x0) ( %( ))- Replacing gr51 by grR-{x0, y0) if necessary, we may assume that (xo,yo) = (0,0). Let us first show that Vi£]0,l[, VU,U' £Kx • tc\3l(U)c 31 {t{U + U')). (1.9) 2 Fix t G ]0,1[ and take tn := t for n > 1. Of course, lim£„ • • • t\ = t. Consider U' G Kx- There exists a base of neighborhoods (t/„)neN C Kx such that U\ +U\ C U' and Un+i + Un+i C Un for every n € N. Let -1 Un := (1 - i„) i„ • --tyUn- For every n G N there exists Vj( 6 Ky such that V^ C cl#([/'„). Consider V„ := ^(l - t„)Vn. Let £/ G Kx and j/ G cl %{U); we intend to show that ty G 31 (*(£/ + £/'))• We construct a sequence x = XQ G U, X\ G t/i, ..., xn G J7n,... with the property: VnGN : tn .. .hy G clK (i„ .. .tx (x + ••• + xn-i +Un)). (1.10) Since (t/ - Vi) n 3?([7) ^ 0, there exist yx G Vi and x G J7 such that y — yi G 3£(a:); hence tij/ = *i(y - yi) + (i - *0 ((i - hyhm) G t&ix) + (l - W C ti3i(a;) + (1 - ti)clK(^) C c\5t{h(x + Ui)). Suppose that we already have x £ U, xi £ Ui,... ,xn-i G £/n-i with the desired property. Since (i„ ... tiy - Vn+i) n 31 (i„ ... h(x + • • • + x„_i + Un)) + 0, an there exist yn+i G Vn+i d xn G C/n such that tn---hy-yn+i G 3i(tn...ti(H Fx„-i + xn)). Therefore _ tn+l •••hy = tn+l(tn- --hy — 2/n+l) + (1 - £ tn+i$.(tn...ti(x-{ ha;„_! +xn)) + (1 -tn+1)V„+1 C i„+i^(*n .. .ti(i + • • • + a;n-i + a;„)) + (1 - t„+i) d3l(t/;+1) C cl3t(tn+i ...h(x-\ hx„_i +xn + Un+i)). So the desired sequence is obtained. Since xn+\ H h x„+p G Un+i + • • • + x s Un+P C Un for all n,p € N, the series 2n>i ™ ' convergent. Denote by x1 its sum. Since ii + - + i»Eii + f/i and Ui is closed we have that x1 Gxi+Ux C U'. 22 Preliminary results on functional analysis Let now U" E Nx and V" E Ny be arbitrary. There exists n E N such that tn ...h(x-\ \-xn-i+Un) C t(x+x')+U" andtn .. .hy G ty+V". By Eq. (1.10) we obtain that tn ... txy E cl!R(t(x + x') + U"). It follows that (ty + V")nJi(t{x + x') + U") ^ Hi, and so there exist x" E U" and y" G V" such that ty + y" E $.(t(x + x')+x"), i.e. (t(x + x'),ty) + (x",y") € gr#. It follows that (t{x + x'), ty) E cl (gr 3V) = grft. Therefore ty E ft (t(U + U')). To complete the proof, let U E Nx(0). Then there exists U' E Nx such that [/' + [/' C 1U. From Eq. (1.9) we obtain that |cl#(E/') C 3? (|([/' + [/')) C #(*/). Since 0 E C\ue^x{xo) int (cl3i(U)), we have that 0G int£([/). ° D Note that Lemma 1.3.3 is a particular case of Lemma 1.3.1 when Y is first countable; otherwise these results are independent. Corollary 1.3.4 Let Y be a first countable topological vector space and C CY be an ideally convex set. Then intC = int(clC). Proof. Consider X an arbitrary Frechet space (for example X = R) and ft : X =t Y denned by IR(0) = C, R(x) = 0 for x ^ 0. Of course condition (Hwir) is satisfied. Taking y0 E int(clC) and xo = 0 we have that 2/o G f)ue^x(xo)int (cl3i(C7))' and so 2/o € int C. O Theorem 1.3.5 (Simons) Let X and Y be first countable. Assume that X is a locally convex space, "R : X =$ Y satisfies condition (Hwa;), yo E lb _1 (ImJi) and x0 E 3l (j/o)- Then yQ E intaff(ims) R(U) for every U E i6 i6 Nx(z0). In particular (ImIR) = rint(Imft) if (ImK) ^ 0. Proof. Once again we may consider that (xo,yo) = (0,0); so Y0 := aff(ImCR) — lin(Im3i). Replacing, if necessary, Y by YQ, we may suppose c that Y is barreled and 0 G (ImIR)\ Let U E N x; since grft is a convex set and R(U) = Pry (gr ft n U x Y), %(U) is convex, too. "R{U) is also absorb ing. Indeed, let y E Y. Because Imft is absorbing, there exists A > 0 such that Xy E Imft. Therefore there exists x E X such that (x,Xy) E grft. Since U is absorbing, there exists fi G]0,1[ such that fix E U. As grft is convex we have that (fj,x,fj,Xy) = fx(x,Xy) + (1 — /x)(0,0) G grft, whence /xAj/ G R(U). Therefore $.(U) is also absorbing. It follows that cl (£([/)) is an absorbing, closed and convex subset of the barreled space Y. Therefore 0 G int (clft(C/)). Using Lemma 1.3.1 we obtain that 0 G intft(Z7) for every neighborhood U of 0 G X. The last part is an immediate consequence of what was obtained above. • Open mapping theorems 23 Corollary 1.3.6 Let X be a Frechet space, Y be first countable and 51 : X =$ Y be li-convex. Assume that yo € t6(Im!R) and XQ 6 5i~1(yo)- n b Then y0 £ intaff(im3j) 5l(U) for all U E Nx(zo)- I particular * (Im5l) = rint(Im^) provided i6(ImK) ^ 0. Proof. There exist a Frechet space Z and an ideally convex multifunction S : Z x X =4 Y such that grIR = Prxxy(grS)- Then § verifies condition (Hw(z,i)) by Proposition 1.2.6 (ii). Of course, there exists ZQ 6 Z such that yo £ S(zo,zo)- Since ImS = Imft, by the preceding theorem, j/0 € intaff(imX) #(E0 = intaff(imK) HZ x ^) for every U G Kx(^o)- • Theorem 1.3.7 (Ursescu) Let X be a complete semi-metrizable locally convex space and 51 : X =£ Y be a closed convex multifunction. Assume t6 1 that yo £ (ImCR) and xo € 5l~ (yo)- Then yo € intaff(Im^) 5l(U) for every U £ Xj(xo). In particular i6(Im^) = rint(ImK) if ib{Im 51) ^ 0. Proof. If Y is first countable it is obvious that the conclusion follows from Simons' theorem. Otherwise the proof is exactly the same as that of Simons' theorem but using Lemma 1.3.3 instead of Lemma 1.3.1. • An immediate consequence of Theorem 1.3.5 is the following corollary. Corollary 1.3.8 Let Y be a first countable barreled space and C CY be a lower ideally convex set. Then Cl = intC. Proof. Let yo G Cl. There exist a Frechet space X and an ideally convex multifunction 51: X =3 Y such that C = Im 51. The conclusion follows from Theorem 1.3.5 taking again U — X. O In fact the conclusion of the above corollary holds if C is the projection on Y of a subset A of X x Y with (a') X is a first countable locally con vex space and A satisfies condition (Hwx) or (b') X is a semi-metrizable complete locally convex space and A is closed and convex. Putting together Corollaries 1.3.4 and 1.3.8 we get the next result. Corollary 1.3.9 Let Y be a first countable barreled space and C C Y be an ideally convex set. Then Cl = intC = int(clC) = (clC)\ • In normed spaces the following inversion mapping theorem holds. Theorem 1.3.10 (Robinson) Let (X, ||-||) and (Y, ||-||) be normed vector spaces, 51 : X =S Y be a convex multifunction and (xo,yo) € gr^- Assume 24 Preliminary results on functional analysis that y0 + rjUy C 3\{XQ + pUx) for some n,p > 0. Then 1 dfc.K- ^)) < P+\\x-x0\\ ,d( ^(a.)) VarGX, Vy£y0 + r,BY. v-\\y-yo\\ Proof. Replacing gr# by gr# — (xo,yo), we may assume that (xo, yo) = (0,0). Let x E X and y G nBy. The conclusion is obvious if x £ domIR or y € &(z), so suppose that neither is true. Choose 6 > 0 and find ye € 3£(:c) such that 0 < \\ye - y\\ < d[y, R(x)) + 6; define a := n - \\y\\ > 0 and take e G ]0, a[. Consider 1 ye :=y + {a-e)\\y-ye\\~ (y - ye); thus ||y£|| < \\y\\ + (a — s) =n — e, and so ye G nBy- Therefore there exists -1 xe G pUx with y£ G R(x€). Define A := \\y -yg\\(a- e + \\y - y^l) G ]0,1[. Then 2/ = (1 - X)ye + Xys G (1 - A)tt(a:) + A#(x£) C E((l - A)z + Aa;£). _1 1 Thus (1 - A)z + Xx£ G 3£ (j/), whence d (x,^- ^)) < A||x-xe||. As ll^ — ^ell < \\x\\ + ll^ell < P + INI and A < (a — e)_1 ||y — yg\\, we obtain that 1 d (x^iy)) <(P+ \\x\\) (a - e)' (d(y, X(x)) + 9). Letting 6, e —> 0, we obtain the conclusion. • Combining the preceding result and Lemma 1.3.2 we obtain the follow ing important result in normed spaces. The implication (i) =$> (ii) is met in the literature as the Robinson-Ursescu theorem. Theorem 1.3.11 Let (X, ||-||), (Y, ||-||) be two normed vector spaces and 31 : X zt7 be a multifunction. Suppose that Y is a barreled space, gr# verifies condition (Hwx) and (xo,yo) G gr3i. Then the following conditions are equivalent: (i) yoeQmX)'; (ii) 2/0 e int:R(xo + Ux)\ (iii) 3n > 0, VA G [0,1] : 2/0 + XnUy C IR(a;o + At/X); (iv) 37,77 > 0, Vz G z0 + r?t/X; Vy G y0 + nUy : d (^ST^y)) < 7-d(y,K(a;)); (v) 3r?>0,VyGyo + r?[/x, VxGX : d(x,-R-\y)) < ^Efff • d(y,3i(i)). Open mapping theorems 25 Proof. The implications (hi) =>• (ii) =>• (i) are obvious. The implication (i) =$> (ii) follows immediately from Simons' theorem, while the implication (ii) =$> (v) is given by the preceding theorem. (iv) => (iii) Let 7' > 7; we may assume that 777' < 1, because, in the contrary case, we replace 77 by 77' := I/7' < 77. Let y £ yo + A?7?7y, y 7^ j/o, 1 withAe]0,l]. Thend(a;o,^- (y)) < 7 • d{y,3L(x0)) < 7' ||y - yQ\\. Hence there exists x € 5l~1(y) such that ||x —io|| < 7'|l2/~2/o|| < 7'Ar/ < 77. Therefore y0 + A^C/y C R(x0 + \UX)- (v) => (iv) Taking a; € XQ + §t/x, 2/ G j/o + f^y' we obtain that d (x, ft"1 (y)) (Tx0,x0) G Y x X. Therefore 1 VVeNHTio) : R(V)=T- (V)£Nx(x0), which means that T is continuous at XQ. • Corollary 1.3.13 (Banach-Steinhaus) Let X, Y be Frechet spaces and T : X —> Y be a bijective linear operator. Then T is continuous if and only if T_1 is continuous; in particular, if T is continuous then T is an isomorphism of Frechet spaces. Proof. Apply the closed graph theorem for T and T-1, respectively. D Theorem 1.3.14 (open mapping theorem) Let X, Y be Frechet spaces and T G L(X,Y) be onto. Then T is open. Proof. Of course, T is a closed convex relation and Im T = Y. Let D C X be open and take 7/0 G T{D)\ there exists XQ G D such that 7/0 = Tx0. Applying the Ursescu theorem for this point, since D is a neighborhood of XQ, we have that T(D) is a neighborhood of T/0. Therefore T(D) is open.D 26 Preliminary results on functional analysis An interesting and useful result is the following. Corollary 1.3.15 Let X, Y be Frechet spaces, A C X and T : X ->• Y be a continuous linear operator. Suppose that ImT is closed. Then T(A) is closed if and only if A + ker T is closed. Proof. Replacing, if necessary, F by ImT and T by T" : X -> ImT, T'{x) := T(x) for x G X, we may suppose that T is onto. Consider T : Xj kerT —> Y, T(x) := T(x), where x is the class of x. It is easy to verify that T is well defined, linear and bijective. Let q : X —> M. be a continuous semi-norm; since T is continuous, p := qoT is a continuous semi-norm, too. For all x € X and u € ker T we have that ( X/keiT, ir(x) := x, we obtain that T(A) is closed & n(A) - A is closed <$• IT'1 (A) = A + ker T is closed, and so the conclusion holds. • As an application of the Simons and Ursescu theorems we give the fol lowing two interesting results, useful in studying optimal control problems. We recall that a process is a multifunction C : X => Y whose graph is a cone; when the graph of the process C is convex or closed one says that e is a convex process or closed process, respectively. The adjoint of the process 6 is the w*-closed convex process 6* : Y* =3 X* whose graph is the set {(y*,x*) £ Y* x X* | (-x*,y*) € (gre)+). Theorem 1.3.16 Let X, Y, Z be Banach spaces, 6 : X =$Y be an ideally convex process and T G £(•£, Y). Consider the following statements: (i) ImT dm6; (ii) 3Pl>0, V(»*,*•) GgrC* : \\T*y*\\ < Pi\\x*\\; (hi) 3P2>o : T{UZ) c P2e(uxy, (iv) 3p3>0 : T(Uz)Cp3c\(e(Ux)). Then (i) «• (hi), (ii) •£• (iv) with p\ = p3 and (hi) => (iv) with p3 = p2. Moreover, ifT is onto then (iv) => (hi) with (any) p2 > p3. Proof. The implications (hi) =$• (i) and (iii) => (iv) (with p3 = p2) are obvious. Open mapping theorems 27 (i) => (hi) Let 3? := T~1oQ; one obtains immediately that 3J is an ideally convex process with ImIR = Z. Applying the Simons theorem (Theorem 1.3.5) for (a;o,2/o) = (0)0); there exists p > 0 such that pUz C 3£(t/x), which means that pT(Uz) C G(Ux)- Taking P2 := p"1, (hi) holds. (iv) => (ii) Let (y*,x*) G grC*, i.e. (-z*,y*) 6 (grC)+. Then ||TVII= sup (z,-TV>= sup (Tz,-y*)< sup fo.-y*) ||z|| = p3 sup (y, -y*) < p3 sup{(a;, -x*) \ (x,y) G grC, ||a;|| < 1} yee(ux) %T*T) = (Tz,y*) < A < (p3y,y*) Vy G G(UX). It follows that A < 0, and so we can take A = -p3. Hence —p3 > (z, T*y*) > -||r*y*ll and -1 < (y,^) = (x,0) + (y,y*) for (x,y) G grC with x G Ux, whence ||T*y*|| > p3 and (0, y*) G (gr 6 n Ux x 7)° = «;* - cl ((gr 6)+ + tf*. x {0}) = (gre)+ + ?7x' x{0} because Ux* x {0} is W*-compact (we have used the Alaoglu-Bourbaki theorem). Therefore there exists x* G Ux* such that (y*,x*) G gr(2* ( T is open (see Theorem 1.3.14). It follows that int(T(Uz)) = T{BZ). But G(Ux) is cs-closed. Indeed, by Proposition 1.2.2, A := grCnUx x Y is cs-closed, and so, X being complete, A satisfies condition (Hz) (see Propo sition 1.2.6(h)); by Proposition 1.2.6(v) we have that Pry(A) = G(Ux) is cs-closed. Using Corollary 1.3.4 we obtain that intC([/x) = iat(c\G(Ux))- So, from (iv) we obtain that T(BZ) C p3Q(Ux), and so T{UZ) C p A similar result holds in dual spaces. 28 Preliminary results on functional analysis Theorem 1.3.17 Let X,Y,Z be normed spaces, T e £>{X,Y) and 6 : X =$ Z be a convex process. Consider the following statements: (i) ImT'Clme*; (ii) 3p>0, V(x,z) SgrC : ||Ta;|| < p||;z||; (iii) 3p>0 : T*(UY*)CPe*(Uz>). Then (i) O- (iii) and (ii) O- (iii) with the same p. Proof. The equivalence of (i) and (iii) follows from the equivalence of (i) and (iii) of the preceding theorem; just apply it for the Banach spaces X*, Y*, Z*, the continuous linear operator T* and the closed convex process e*. (iii) =>• (ii) The proof follows the same lines as the proof of (iv) =£• (ii) in the preceding theorem, so we omit it. (ii) => (iii) Even if the proof is similar to (ii) =>• (iv) in the preceding theorem, we give the details. So, suppose that (iii) does not hold and take y* e Uy* such that T*y* £ pG*(Uz>). We may assume that y* £ Sx- (otherwise replace y* by ||2/*||_1y*). Using the fact that Uz* is w*-compact and gre* is u>*-closed it follows easily that G*(Uz*) is uAclosed; being also convex, we can apply Theorem 1.1.5 in (X*,w*). Therefore there exist x £ X and A G E such that (Tx,y*) = {x,T*y*) > A > (x,px*) Vx* € e*{Uz>). Hence A > 0, and we can take X = p. Hence ||Tx|| > p and 1 > (x, x*) for every x* € e*(Uz>), i.e. -1 < (x,x*) + (0,z*) for (x*,z*) 6 (grC)+ with z* £ Uz* • It follows that (af,0) € ((grQ+fllx Uz.)° = cl ((gre)++ + {0} x UZ) = cl(gve+{0}xUz). Hence there exist ((xn,zn)) C grC and (z'n) C Uz such that (xn) -> x and (wn) := (zn + z'n) ->• 0. It follows that (||Ta;„||) ->• \\Tx\\ and \\zn\\ = \\wn — z'n\\ < 1 + ||iwn|| for every n, whence plimsup \\zn\\ < p < \\Tx\\ < liminf ||Ta;„||. Therefore there exists n0 such that ||Ta;n|| > p||z„|| for n >n0, which proves that (ii) does not hold for the same p. O Variational principles 29 1.4 Variational Principles A very important result in nonlinear analysis is the "Ekeland variational principle." Theorem 1.4.1 (Ekeland) Let {X,d) be a complete metric space and f : X —» M. be a proper lower semicontinuous and lower bounded function. Then for every xo G dom / and e > 0 there exists xe G X such that f(xE) < f(x0) -ed(xQ,xE) and f{xe) Multiplying the last relation by e, then adding all three relations we obtain that f(z) + ed(x,z) < f(x), i.e. z G F(x). Since / is bounded from below, s0 := inf{/(x) | x G F(xo)} G E; take X\ G F(xo) such that /(^i) < SQ + 2~1. Then consider si := inf{/(z) | x G 2 F(xi)} G E and take x2 G F(a;i) such that ffa) < s\ + 2~ . Continuing in this way we obtain the sequences (sn)n>o C M. and (xn)n>o C X such that - -1 sn = inf{/(a;) | x G F(xn)}, xn+1 G F(xn), f{xn+1) < sn + 2 " for all n > 0. Because i^(a;n+i) C F(xn), s„+i > s„ for n > 0. Moreover, as xn+i G F(i„), n ed(a;ri+i,a;n) < f(xn) - f(xn+1) < f{xn) - sn < f(xn) - s„_i < 2~ 1 1 n for n > 1, whence d(x„+p,a;„) < e~ 2 ~ for n,p > 1. This shows that (xn) is a Cauchy sequence, and so (xn) converges to some xs G X. Since 30 Preliminary results on functional analysis xm G F{xn) for m > n, we have that xE € clF(a;n) = F(xn) for every n > 0. In particular xE £ F{XQ). Let a; 6 F(ar£); then x € F(xn) for every n n > 0, and so, as above, ed(x,xn) < f(x„) — f(x) < f(x„) - sn < 2~ , which shows that (xn) —>• x. As the limit is unique, we get x = xe, which shows that F{xe) = {xe}. The proof is complete. • In applications one often uses the following variant of the Ekeland variational principle. In the sequel by infx /, or simply inf/, we mean mi{f(x) | x € X}, where / : X -> I. Corollary 1.4.2 Let (X,d) be a complete metric space and f : X -» R be a bounded below lower semicontinuous proper function. Let also e > 0 and xo S dom/ be such that f(xo) < mix f + £• Then for every A > 0 there exists x\ S X such that f(x\) < f(x0), d(xx,x0) < A and f(xx) < f{x) + eX-idfaxx) Va; £ X \ {xx}. Proof. Applying the preceding theorem for xo and eA_1, we get x\ € X satisfying the second relation of the conclusion and f(x\)+e\~1d(xo,x\) < f(xo). Hence f{x\) < f{xo). Moreover, because f(x0) < inf^ / + £ < f(x\) + e, we get also that d{x\, XQ) < A. D A good compromise is obtained by taking A = -Jz in the preceding result. An interesting application of the Ekeland theorem is the following result. Corollary 1.4.3 Let (X, ||-||) be a Banach space and f : X -> R be a lower semicontinuous function. Assume that f is Gateaux differ•entiable and bounded from below. If (xn)n^ is a minimizing sequence for f, i.e. x (f( n)) -> infjt /, then there exists a minimizing sequence (xn) for f such that (\\xn -xn\\) -> 0 and (||V/(x„)||) -» 0. Proof. Consider en := f(xn) — infx f £ K+- If f{xn) = infx f we take xn := xn; otherwise, applying the preceding corollary for xn, en and A = y/e„~ we get xn e X such that f(x„) < f(xn), \\x - xn\\ < ^fe^ and f(xn) tu) — f{xn)). Taking the limit for t —¥ 0 we get — y/e^\\u\\ < (u,S7f(xn)) for all u £ X. It follows that ||V/(z„)|| < y/e^. The conclusion follows. • The next result is met in the literature as the smooth variational prin ciple. This name is due to the fact that, at least in Hilbert spaces, the perturbation function is smooth (for p > 1). The result will be stated in a Banach space (X, ||-||) even if it holds in any complete metric space (with the same proof). Before stating it let us observe that, when (un)n>o C X an is bounded, (nn)n>o C M+ is such that Yln>o ^n — 1 ^ P £ [1J °°[> the function n Un L11 Qp : X -+ E, Qp(x) := ^2n>0^ ^ ~ ^ ' ( ) is well defined and Lipschitz on bounded sets; in particular 0P is continuous. Theorem 1.4.4 (Borwein-Preiss) Let (X, ||-||) be a Banach space, f : X —> M. be a proper lower semicontinuous and bounded from below function and p 6 [1, oo[. If xo £ X and e > 0 are such that f(xo) < mix f + £ then for every A > 0 there exists a sequence (wn)n>o C B(xo,X) converging to some u € X such that f(u) < mix f + e, \\xo - u\\ < A and p p f(u) + e\- Qp{u) < f(x) + e\- Qp{x) VxeX, (1.12) where Qp is defined by Eq. (1.11). Proof. Fix A > 0 and consider 7, S, n, /i, 6 > 0 such that 1 1/ f(x0) - mixf < V < 7 < e, H Hence there exists u2 £ X such that f2(u2) < 6f\(u\) + (1 — 0)infx /2. Continuing in this way we obtain a sequence (u„)n>o C X and a sequence of functions (fn)n>o such that n p fn+i(x)=fn(x)+6fi \\x-un\\ Vx£l,Vn>0, (1.14) 32 Preliminary results on functional analysis and fn+l{Un+l) It is clear that /„ < fn+i; taking sn := infj*:/n, (s„)„>o C E is a non- decreasing sequence. Let an := /„(un); because fn+i{un) = fn(un) > infx/n+i, from Eq. (1.15) we obtain that an+\ < an for every n > 0. Using again Eq. (1.15), we get sn < s„+i < an+i < 0an + (1 - 9)sn+i < an, and so a-n+i - sn+i < Qan + (1 - 6)sn+i - sn+i = 6(an - sn+i) < 0(an - s„), whence n an-sn<9 (a0-s0) Vn > 0. (1.16) It follows that lims„ = lima„ € E. Taking x := un+i in Eq. (1.15), we get n p fln > a«+i = fn{un+i) + Sn" \\un+1 - un\f >sn + 6p, ||u„+i - un\\ , which, together with Eqs. (1.13) and (1.16), yields n n Sfi" ||«n+1 - u„f 1 J, It follows that ||un+1 - un\\ < (77/<5) / (6'/^)"/P, whence 1/p n,p 1 1 \\un+m -u„|| < (v/8) (0/l*) {l ~ Wltf ')' Vn,m > 0. Since 0 < #//z < 1, the sequence (un)„>o is Cauchy, and so it converges to some u 6 X. Moreover, the inequality above and (1.13) imply that IK - unll < (V/S)1/P(vhr1/P = (l/S)1/p < (e/Sy/p(l - fi)1^ = A (1.17) for all n,m > 0. In particular (un) C B(xo,X)- Letting m —> co in the above inequality we get \\u — un\\ < A for n > 0, and so \\u — xo\\ < A. n 1 Consider n„ := fi (l - fx) > 0; hence J2n>of n = 1- Let 0P be defined by Eq. (1.11). From Eq. (1.14) we obtain that p n p fn+i{x)=f{x)+e\- Y. k=0^\\x-uk\\ Vn>0, Vxel, Variational principles 33 p and so f(x) + eX Qp(x) — lim/„(a;) > lims„. Using Eq. (1.17) we get p p p f(un) + e\~ Qp(un) = fn(un) + e\~ Y] ^k \\un - uk\\ fikX" = an + e} ak K=n+1 *—/k=n+l for every n > 1. Taking into account the continuity of 0P and the lower p semicontinuity of /, the preceding inequality yields f(u) + eX~ Qp(u) < lima„ = lims„. Therefore (1.12) holds. From Eq. (1.17) we obtain that P 1 -1 P Qp{x0) < Y" ^Un IK - Un\\ < J~] HkjS" = fJ.jS < fJ,X , ^—'n>l ^—'n>l and so, by Eq. (1.12), p f{u) < f(x0) + eX~ ep{x0) < f(x0) +sfi< infx / + 7 + £/* < infx f + £- The proof is complete. D Note that |©i(:r) -Qi(y)\ < \\x — y\\, and so Eq. (1.12) becomes f(u) < /(a;)+£A-1||a;—w|| for every 2; € Xwhenp= 1. So, under the slight stronger condition f(x0) < mix / + £ one recovers the conclusion of Corollary 1.4.2, but the condition f(x\) < f(xo). Although not directly related to what follows, we give the next two dual interesting results which have not been published by their author, C. Ursescu. Theorem 1.4.5 Let (X,d) be a complete metric space and (F„)neN be a sequence of closed subsets of X. Then cl intF dint 18 (U„eN ")= (U„eN^)- d- ) Theorem 1.4.6 Let X be a complete metric space and (Dn)ne^ be a sequence of open subsets of X. Then int ci =intci LI9 (n„6N ^) (n„eN^)- ( ) Note that Theorem 1.4.5 can be obtained immediately from Theorem 1.4.6 by taking Dn = X\Fn (and Theorem 1.4.6 is obtained from Theorem 1.4.5 by taking Fn = X \ Dn). We give only the proof of Theorem 1.4.6. Proof. Since the inclusion "D" in Eq. (1.19) is obvious, we show the converse one. So let x G int (|"|neNclDn) and r > 0; take xi := x. 34 Preliminary results on functional analysis Then there exists ri G]0,r[ such that D{xi,r\) C f]n£Nc\Dn; there fore D(xi,r\) C clD„ for every n £ N. It follows that B(x\,ri) n D\ is an open nonempty set. There exist X2 € X and r2 6]0,ri/2] such that D(x2,r2) C B(xi,n) D £>i. It follows that Dfo,^) C D(a;i,ri) C cl£>2) and so B{x2,r2) fl £>2 is an open nonempty set. Continuing in this way we find the sequences (a?n)neN C X and (rn)n£N C F such that (r„)n£m ->• 0 and D(xn+\,rn+i) C B(x„,r„) n £>„ for every n. In partic ular D(a;n+i,rn+i) C D(xn,r„) for every n. Since X is a complete metric space, using Cantor's theorem, (~}neND(xn,rn) = {x'} for some x' € X. It follows that x' £ £>„ for every n. Since x' 6 D(xi,r{) C B(x,r), we have that B(x,r)nf\n€JiDn. As r > 0 is arbitrary, x £ cl (finer*-^M- Therefore int (f|neNclD„) C cl (f|n€NAi), whence the inclusion "C" in Eq. (1.19) holds, too. • From Theorem 1.4.5 we obtain immediately the famous Baire's theo rems: Theorem 1.4.7 (Baire) Let (X,d) be a complete metric space. Then any countable intersection of dense open subsets of X is dense. Proof. Let A = C\n€N Dn with Dn open and dense for every n £ N. From Eq. (1.19) it follows that X = int(clA), and soc\A = X. • Remind that an intersection of a countable family of open subsets of the topological space (X, r) is called a Gs set. Theorem 1.4.8 (Baire) If (-Fn)ngN is a sequence of closed subsets of the complete metric space (X,d) such that X = \Jne^Fn, then at least one of Fn 's has nonempty interior. Proof. The conclusion is immediate from Eq. (1.18). • 1.5 Exercises Exercise 1.1 (Caratheodory) Let X be a linear space of dimension n6N and let A C X be nonempty. Prove that co A = { Y^ill^ | Mie WT C R+, (xi)ieT^+T C A, Y^ill^ = !} • Exercise 1.2 Let X be a finite dimensional normed space and A C X be a nonempty convex set. Prove that 'A =£ 0, "(cl A) — 'A, cl(*A) = cl A and for every a € 'A and z g cl A we have }a, z[ C lA. Exercises 35 Exercise 1.3 Let X be a separated locally convex space, H C X be a closed hyperplane and A C X be a nonempty convex set. If int# (A O H) ^ 0 and A <£ H prove that int ^4 ^ 0. Moreover, if M C X is a closed affine set with finite codimension and intM{A f~l M) ^ 0, prove that rint A ^ 0. Exercise 1.4 Let X be a separated locally convex space and A, C C X be non empty convex sets with int C ^ 0. Prove that int(A + C) = A + int C. Moreover, if A n int C ^ 0, then cl(A DC) = cl yl l~l cl C. In particular, if C is a convex cone with nonempty interior, then cl C + int C — int C. Exercise 1.5 Let X be a separated locally convex space, Xo C X be a linear subspace, ai,..., ap G X, ipi,..., tpk G X" and ai,..., a* G R, where p, A; G N. Prove that: (a) if Xo is closed and C = {ELi-^ I Vi G L7p : A, > 0}, then X0 + (7 is a closed convex cone. In particular C is a closed convex cone; (b) Xo + {x £ X | Vi 6 1, k : (x, ipi) < at} is a convex closed set. Exercise 1.6 Let (X, (• | •)) be a Hilbert space, xo £ X \ {0} and a G [0,7r/2]. Prove that P(a) := {x £ X | Z(a;, x0) < a} is a closed convex cone, where Z-(x,y) := Arccos " . . Moreover, P(a)° = P(7r/2-a). Exercise 1.7 Let n £ N\ {1}, p > 0 and ai,...,a„ £]0,1[ be such that ai + ... + an = 1. Consider n+1 1 ATp := {(xi,...,x„,xn+i) G R | xi,...,x„ > 0, |xn+i| < px" •• -a;""} • + 1 -1 Prove that (Kp) = .Ry, where p' := (pa" •••a"") . In particular, Kp is a closed convex cone. Exercise 1.8 Prove that P := {(x,t/,z) £R3 | x,z >0, 2xz> t/2} is a closed convex cone and P+ = P. (P is called the "ice cream" cone.) Exercise 1.9 Let X be a normed space, ip G X* \ {0} and 0 < a < \\(p\\. Prove that the set C := {x G X | Exercise 1.10 Let X, Y be topological vector spaces and 31 : X =t Y be a convex multifunction. Assume that there exists x G X such that int 3^(5:) ^ 0. Prove that for every (xo,j/o) G grft with yo G (Imft)1 there exists u G X such that j/o G intK(x) for every x &}xo,u]. 36 Preliminary results on functional analysis Exercise 1.11 Let X, Y be two separated locally convex spaces and 6 : X =4 Y be a multifunction whose graph is a closed linear subspace of X x Y. Prove that domC is dense if and only if C*(j/*) is a singleton for every y* G dome*. In particular, dom (2 is dense and C(x) is a singleton for every x G dom G if and only if domC* is w*-dense and C*(y*) is a singleton for every y* € dom6*. Exercise 1.12 Let (X, d) be a metric space. Prove that (X, d) is complete if for any Lipschitz function / : X —• R+ there exists x € X such that f(x) < f(x) + d(x,x) for every x £ X. This shows that the completeness assumption in Ekeland's variational principle is essential. Exercise 1.13 Let (X, d) be a complete metric space and / : X —)• R be a proper lsc and lower bounded function. Suppose that for every x G X such that f(x) > inf /, there exists x G X\{x} such that f(x) + d(x,x) < f(x). Prove that argmin / ^ 0 and d(x, argmin /) < f(x) — inf / for every x E X. Exercise 1.14 Let (X, d) be a complete metric space, / : X —• R be a proper lsc lower bounded function and 3£ : X =t X. Assume that for every x G X there exists y G 3l(x) such that d(x,y) + f(y) < f(x). Prove that 31 has fixed points, i.e. there exists xo S X such that xo € 3£(xo)- Exercise 1.15 Let (X, ||||) be a normed space and / : X -> R. Prove that (i) limn^n^oo f(x) = oo if and only if [/ < A] is bounded for every A G R. (ii) Assume that X = Rfc, / is lsc and limnsn^oo f(x) = oo (i.e. f is coercive). Prove that there exists x G R* such that f(x) < f(x) for every x G Rfc. 1.6 Bibliographical Notes Section 1.1: For the notions and results on topology and topological vector spaces not recalled in this section one can consult many classical books (see [Kelley (1955); Willard (1971); Bourbaki (1964); Holmes (1975)]). The proofs of the results mentioned in this section can be found in [Holmes (1975)] or [Bourbaki (1964)]. Section 1.2: Ideally convex sets were introduced by Lifsic (1970), cs-closed sets by Jameson (1972), cs-complete sets by Simons (1990), lower cs-closed sets by Amara and Ciligot-Travain (1999), while condition (Hx) was used by Zalinescu (1992b). The properties stated in Proposition 1.2.1 are mentioned in [Kusraev and Kutateladze (1995)]. All the results concerning lcs-closed sets, as well as Proposi tion 1.2.2 (for cs-closed sets), are from [Amara and Ciligot-Travain (1999)]. The set *CA, introduced by Zalinescu (1987), is also introduced by Jeyakumar and Wolkowicz (1992) under the name of strong quasi relative interior of A. The notation lbA was introduced in [ZalinescuN(1992b)] but the condition 0 G lbA was intensively used by Simons (1990) and Zalinescu (1992b). The quasi relative interior was introduced by Borwein and Lewis (1992); they stated the proper ties mentioned in Proposition 1.2.7 in locally convex spaces. Proposition 1.2.9 is Bibliographical notes 37 stated in [Borwein and Lewis (1992)] for A a cs-closed set and X a Frechet space. Joly and Laurent (1971) introduced the notion of united sets, but Proposition 1.2.8 is mainly established by Moussaoui and Voile (1997). Section 1.3: Lemma 1.3.1 is proved by Amara and Ciligot-Travain (1999) for X a metrizable lcs, Y a Frechet space and ft cs-closed, while the statement and proof of Lemma 1.3.3 are those of Ursescu (1975); Corollary 1.3.4 (for Banach spaces) is due to Lifsic (1970). The statement of Theorem 1.3.5 is very close to that of [Kusraev and Kutateladze (1995), Th. 3.1.18]; when X, Y are Frechet spaces our statement is slightly more general. For X, Y metrizable locally convex spaces and CR. satisfying (Ha;) Theorem 1.3.5 is equivalent to the open mapping theorem of Simons (1990). Theorem 1.3.7 is obtained by Ursescu (1975) for Y0 := aff(Imft) = Y. It is established in [Robinson (1976)] for X, Y Banach spaces and Yo = Y; in this case the above result is met in the literature under the name of Robinson-Ursescu theorem. Corollary 1.3.8 was obtained by Amara and Ciligot-Travain (1999) for lcs-closed sets. Theorem 1.3.10 is due to Robinson (1976), while Theorem 1.3.11 can be found in [Li and Singer (1998)]. Theorems 1.3.16 and 1.3.17 are due to Carja (1989) (with different proofs). Section 1.4: The Ekeland variational principle [Ekeland (1974)] is met in the literature, generally, in the form given in Corollary 1.4.2; the statement and proof of Theorem 1.4.1 are from [Aubin and Frankowska (1990)]. The statement and proof of the Borwein-Preiss variational principle are from [Borwein and Preiss (1987)]. Theorems 1.4.5 and 1.4.6 can be found in the recent book [Carja (1998)]. Exercises: The exercises are generally known results or auxiliary results from various papers. So Exercise 1.1 is the famous Caratheodory theorem, Exercise 1.3 is from [Joly and Laurent (1971)], Exercise 1.5 is from [Laurent (1972)], Exercise 1.7 (for n = 2 and p = p) is from [Bauschke et al. (2000)], Exercise 1.10 is stated in [Li and Singer (1998)] in Banach spaces in a slightly weaker form, Exercise 1.12 is the main result of [Sullivan (1981)], Exercise 1.13 is from [Hamel (1994)], Exercise 1.14 is the known Caristi's fixed point theorem (see [Caristi (1976)]). This page is intentionally left blank Chapter 2 Convex Analysis in Locally Convex Spaces 2.1 Convex Functions We begin this section by introducing some notions and notations. For a function / : X —¥ E and Agl consider: dom/ = {x e x | f(x) < +00}, epi/ = {(x,t) eX x E I f(x) < t}, epis/ = {(x,t)\f(x) Var.l/GX, VAg[0,l] : /(Arc + (1 - \)y) < \f(x) + (1 - X)f(y), (2.1) with the conventions: (+oo) + (—00) = +00, 0-(+oo) = +00, 0-(—00) = 0. Note that when x = y, or A £ {0,1}, or {a;, y} (£. dom /, the inequality (2.1) is automatically satisfied. Therefore the function / is convex if Vx,j/edom/, x^,VAG]0,l[: /(Aar + (l-\)y) < \f(x) + (1 -\)f(y). (2.2) 39 40 Convex analysis in locally convex spaces If relation (2.2) holds with " < " instead of " < " we say that / is strictly convex. Similarly, we say that / is (strictly) concave if —/ is (strictly) convex. Since every property of convex functions can be transposed easily to concave functions, in the sequel we consider, practically, only convex functions. To avoid multiplication with 0, taking into account the above remark, we shall limit ourselves to A G ]0,1[ in the sequel. In the following theorem we establish some characterizations of convex functions. Theorem 2.1.1 Let f : X —>• E. The following statements are equivalent: (i) / is (strictly) convex; (ii) the functions tpxy : E —)• R, (px,y(t) := f((l — t)x+ty), are (strictly) convex for all x, y G X (x ^ y); (hi) dom f is a convex set and Vx,yedomf, VAe]0,l[: f(\x + (1 - X)y) < Xf(x) + (1 - X)f(y) (the inequality being strict for x ^ y); (iv) Vn€ N, \/xi,...,xn E X, VAi,...,A„ G]0,1[, Ai+--- + A„ = 1 : f(XlXl + • • • + Xnxn) < X1f{x1) + ••• + Xnf(xn) (2.3) (the inequality being strict when Xi,..., xn are not all equal); (v) epi f is a convex subset of X x E; (vi) epis / is a convex subset of X x E. Proof. The implications (i) => (ii), (ii)=^(i), (i) =>• (hi), (iii) =4> (i) and (iv) => (i) are obvious. (i) =£• (v) Let {xi,ti),(x2,t2) 6 epi/ and A e]0,1[. Then x\,x2 G dom/, f(xi) < ti and f(x2) < t2. Since / is convex, from Eq. (2.1) we obtain that f{Xxx + (1 - X)x2) < Xf{Xl) + (1 - X)f(x2) < Xtx + (1 - X)t2, whence X{xi,t\) + (1 - X)(x2,t2) € epi/. Therefore epi/ is convex. (v) =^ (iv) Let k 6 N, A!,...,Afc (2.3) is verified. Finally, suppose that /(x,) < oo for every i and that there exists io with f(xi0) = — oo; consider i; G E such that (x;,£i) 6 epi/ for every i ^ io', since (ij0, —n) € epi/ for every n 6 N, we have /(AiXiH hAfcXfc) < AitiH hAj0_i£j0_i+Ai0(-n)+Ai0+iiio+i+.. .+\ktk for n G N. Letting n -»• oo in the above inequality we get f (J2i=i^ixi) = —oo; hence Eq. (2.3) is verified. The proofs for (i) => (vi) and (vi) => (iv) are similar to those of (i) =>• (v) and (v) => (iv), respectively. The proof for the "strictly convex" case is similar. • Note that in (v) and (vi) there are no counterparts corresponding to / strictly convex. Proposition 2.1.2 If f : X —> E is sublinear then f is convex. Moreover, f is sublinear if and only if epi/ is a convex cone with (0, —1) £ epi/. Proof. Let / be sublinear. It is obvious that / is convex, and so epi/ is a convex set which contains (0,0). If (x,t) G epi/ and A > 0 then f(Xx) = \f(x) < Xt, and so X(x,t) G epi/; hence epi/ is a convex cone. Since /(0) = 0 > -1, we have that (0, -1) g epi/. Assume now that epi/ is a convex cone with (0,-1) ^ epi/. It is immediate that / is convex (epi/ being convex) and f(Xx) = Xf(x) for A > 0 and x G X. So, for x,y E X we have that /(a; + y) = 2/(|x + |y) < fix) + fiy)- If /(0) < 0 then (0, -t) € epi/ for some t > 0, whence the contradiction (0, —1) 6 epi/. Therefore /(0) = 0, and so / is sublinear. • The indicator function of the subset A of X is 0 if x G A, i :X->R, i (x) := j A A +oo if x e X\A. Note that dom iA = A and epi t,A = A x R+. From the preceding theorem we obtain that LA is convex if and only if A is convex. If / is convex the sets [/ < A] and [/ < A] are convex for every A G E. The converse is generally false. A function / with the property that "[/ < A] is convex for every A G E" is said to be quasi-convex. So, / : X -» E is quasi-convex if and only if Vz,j/GX, VAG[0,1] : /((l - A)* + Ay) < max{/(a;),/(i/)}. 42 Convex analysis in locally convex spaces The notion of convex function is extended naturally to mappings with values in ordered linear spaces. Let Y be a real linear space and Q C Y be a convex cone; Q generates an order relation on Y: y\ < 2/2 (or 2/1 Va;,2/eX, VAe]0,l[: H{Xx + (1 - X)y) < XH{x) + (1 - X)H{y). The operator H : X ->• F U {-00} is Q-concave if -iJ : X -» F* is Q-convex. If A £ L(X, Y) then ^4 is Q-convex for every convex cone Q C Y. As in the case Y = E, the domain and the epigraph are defined by: domi? := {x £ X | #(2;) < 00} andepii? := {(x, y) eX xY \ H(x) < y}. The characterizations of convex functions given in Theorem 2.1.1 are valid for a Q-convex operator. A function / : (Y, Q) —> E is Q-increasing if 2/1 Theorem 2.1.3 Let X,Y be linear spaces and Q CY be a convex cone. (i) // fi : X —> E is convex for every i € / (/ ^ 0) then supigj fi is convex. Moreover, epi(supi€//,) = P|i6/epi/j. (ii) // /1, /2 : X -> E are convex and X £ E+, tften /1 + fi and A/i are convex, where 0 • /1 := tdom/i • Moreover, dom(/i -1-/2) = dom fi n dom fi, dom(A/i) = dom /1. (hi) If fn : X —t M. is convex for every n £ N and f : X —> E is such that f(x) = limsupn_>0O fn(x) for every x £ X, then f is convex. Convex functions 43 (iv) If A C X xR is a convex set then the function ipA is convex, where ipA:X^R, h:Y^% h(y):=mix€XF(x,y). (2.5) (vi) Let g : Y —> E be a convex function. If H : X —> Y' is Q-convex and g is Q-increasing, then g o H is convex; this conclusion holds also if H : X —> Y U {—oo} is Q-concave and g is Q-decreasing. In particular, if A 6 L(X, Y) then go A is convex. (vii) Let f : X —>• E, g :Y —> K 6e proper convex functions, and let $,V:XxY->R, {x,y):=f(x) + g(y), ¥(x,y) := f(x)V g(y). Then $ and \& are convex and proper. Moreover, dom$ = dom* = dom / x dom g, inf $ = inf / + inf g and inf $ = inf / V inf g. (viii) If f : X —> R is convex and A G L(X, Y) then the function Af:Y^R, (Af)(y):=mi{f(x)\Ax = y}, is convex. Moreover dom(Af) = A(domf) and inf Af = inf/. (ix) If /i, f2 : X —> R are convex and proper, their convolution and max-convolution, defined by f1Df2 : X -> K, (/iD/2)(i) := inf {/i(a;i) + /2(z2) | an + x2 = x} , /1O/2 : X -> E, (f10f2){x) := inf {/i(n) V /2(z2) I Zi + *2 = z} , are convex. Moreover dom(/1D/2) = dom^O./^) = dom/x + dom/2, inf fiDfi = inf /1 + inf /2, inf f10f2 = inf /1 V inf f2, epi4(/iD/2) = epis /1 + epi5 /2 (2.6) and VA 6 E : [AO/2 < A] = [A < A] + [f2 < A]. (2.7) Proof, (i) It is clear that epi(supiejr fi) = f)ieIepifi- Since fi is convex for every i, using Theorem 2.1.1, we have that epi/i is convex, and so epi(supi€/ fi) is convex. The conclusion follows using again Theorem 2.1.1. (ii) is immediate. 44 Convex analysis in locally convex spaces (iii) Let A s]0,1[ and x,y € dom/. Since limsup/„(x), limsup/„(y) < oo, there exists n0 £ N such that fn(x), fn(y) < oo for every n > no- Since /„ is convex, we have fn(Xx + (1 - X)y) < \fn(x) + (1 - X)fn(y). Taking the limit superior we obtain f(Xx + (1 - X)y) < Xf(x) + (1 - X)f(y). Therefore / is convex. (iv) Let (xi,ti), (x2,t2) £ episipA and A £]0,1[. Then there exist si,S2 £ ffi such that (xi,si),(x2,s2) € ^4 and si < £i, S2 < t2- Since A is convex, A(:ci,si) + (1 - A)(z2,S2) = {Xx\ + (1 - A)x2,Asi + (1 - A)s2) £ A. Therefore, (PA(XXI + (1 — A)x2) < Xsi + (1 — X)s2 < Aii + (1 — A)i2, and so X(xi,ti) + (1 — A)(a;2,t2) £ epis y>A- Hence ip^ is convex. (v) Note that epish = PryxR(episF), dom h = Pry (dom F). (2.8) The conclusion follows from Theorem 2.1.1(vi). (vi) We observe that (/ o H)(x) = infy6y F(x,y), where F(x,y) := g(y) + t,ep\H(x,y), because / is (J-increasing. Since obviously F is convex, by (v) we obtain that (/ o H) is convex, too. The other case is proved similarly. The second part is immediate taking into account that A is {0}-convex and g is {0}-increasing. (vii) One obtains immediately that dom $ = dom $ = dom / x dom g and that $ and \? are convex. The formulas for inf $ and inf * are well- known. (viii) We have that (Af){y) = mixZX F{x,y), where F(x,y) := f(x) + >-gTA(x,y)- It is obvious that F is convex. From (v) we obtain that Af is convex. As domF = {(x,Ax) \ x £ dom/}, we have that dom(Af) — A(domf). The formula for inf Af is obvious. (ix) Let us consider the functions $, \P : X x X -y R defined by *(xux2) := fiixx) + f2(x2), *(zi,x2) := h{xi)y f2{x2), and A : X x X -> X, A{x1,x2) := zi + x2. Then (^D^)^) = (A$)(x) and (fiQf2)(x) = (A^!)(x). By (v) we have that $ and $ are convex; using (viii) we obtain that /iD/2 and /1O/2 are convex, too. The relations on the Convex functions 45 domains, epigraph and level sets follow by easy verification. The formulas for inf /iD/2 and inf /1O/2 follow immediately from (vii) and (viii). • The formulas (2.6) and (2.7) motivate the use of "epi-sum convolution" and "level-sum convolution" for the convolution and max-convolution, re spectively. The convolution and max-convolution are extended in an ob vious way to a finite number of functions; from Eqs. (2.6) and (2.7) one obtains that these operations are associative. The convex functions which take the value —00 are rather special. Proposition 2.1.4 Let f : X —»• E be a convex function. If there exists XQ € X such that f(xo) = —00 then f{x) = —00 for every x € '(dom/). In particular, if f is sublinear and 0 S '(dom/), then f is proper. Proof. Let x £ '(dom/); since xo £ dom/, there exists /J, > 0 such that y := (1 + fi)x — \ix§ € dom/. Taking A := (p, + 1)_1 e]0,1[, x = (1 — \)XQ + Ay, and so /Or)<(l-A)/(x0) + A/(2,) = -oo. The case of / sublinear is immediate from the first part. • In the sequel we denote by A(X) the class of proper convex functions defined on X. Let now |/CcI and / : C -> M; we say that / is convex if C is convex and Vx,y€C, VA€]0,1[: /(Ax + (1 - X)y) < Xf(x) + (1 - \)f(y). It is easy to see (Exercise!) that the function / above is convex if and only if ?-X^W ?M-( f{x) if XeC> is convex in the sense of the definition given at the beginning of this section. Of course, we can proceed in the opposite direction; so a proper function / : X —> ffi is convex if and only if /|dom/ is convex in the above sense. The consideration of (convex) functions with values in M. has certain advantages, as we shall often see in the sequel. Taking into account the equivalence (i) <£> (ii) of Theorem 2.1.1, it is useful to know properties of convex functions of (only) one variable. The following result collects some properties of such convex functions. 46 Convex analysis in locally convex spaces Theorem 2.1.5 Let f € A(E) be such that int(dom/) ^ 0. (i) Let t\,ti € dom/, t\ < £2. Suppose that there exists XQ €]0,1[ such that /((l - A0)ii + X0t2) = (1 - A0)/(*i) + A0/(£2)- Then VAe[0,l] : /((l-A)t1+At2) = (l-A)/(*i)+A/(*2), (2.9) h-t s . t-t Vt€[ti,t2] : /(*) = r—rf(ti) + i—±m). (2.10) *2-tt2 — t\i 12 — T\ (ii) Let to 6 dom/. The function /( ( o) Vt0 : dom/ \ {i0} -»• K, <*„(*) == *1 f * , t — to is nondecreasing; if f is strictly convex then tpt0 is increasing. (iii) Let to e dom/. Tfte following limits exist: t4.to I — to *>to I — to r.(tt):-ta!StJM.mpmzmet, (2.12) *t*o t — to t re[/-W,/|(to)]nl«vtei : T(t-t0) < f(t) - f{t0). (2.14) /// is strictly convex, in relation (2.14) the inequality is strict for t ^ to. (iv) Lei fi,i2 £ dom/, £1 < t2. Then f\.(h) < f'_{ti); if f is strictly convex then f\_{t\) < /!_(£2). Therefore the functions f'_ and f'+ are non- decreasing on dom/. Furthermore, f is strictly convex if and only if f'_ [resp. f'+] is increasing on int(dom/). (v) The function f is Lipschitz on every compact interval included in int(dom/), and so f is continuous on int(dom/); moreover, for every to E int(dom/) we have: lim/l(i) = lim/;(i) = f_(t0), lim/;(t) = lim/l(0 = f'+ifo). tjlO tjtQ tllQ t^-to Convex functions 47 (vi) The function / is monotone on int(dom/), or there exists to & int(dom/) such that f is nonincreasing on ] — oo,£o] H dom/ and non- decreasing on [io,oo[n dom/. (vii) Let to € dom/ and A 6]0,1[. Then the mapping tp : dom/ —> R, if>(t) := \f(t) — f(to + \(t — to)), is nonincreasing on Ii :=] — oo,to\Hdom/ and nondecreasing on Ir := [io,oo[ndom/. /// is strictly convex then ip is decreasing on I\ and increasing on Ir. Proof. To begin with, let ti,t2,t3 € dom/ be such that t± < t2 < t3. Then t2 = {1 - X)tx + \t3 with A = (t3 - t2)/(t3 - h) € ]0,1[; therefore m)<^-m) + ^m), (2.15) the inequality being strict if / is strictly convex. Subtracting successively f(h), f(h), f(t3) from both members of Eq. (2.15), multiplying then by and t£u> it,-tl)(tl-t2) A> respectively, we obtain m)-m) m)-w) f(tl) - /(*3) < /(*2)-/(*3)> (216) *i — *3 ~ *2 — i3 these inequalities being strict if / is strictly convex. (i) Let ti,t2 € dom/ and Ao G]0, l[be such that /((l - A0)ii + A0i2) = (1 — A0)/(*i) + X0f(t2). Suppose that there exists A e]0,1[ such that /((l - \)ti + Xt2) < (1 - X)f(h) + Xt2; assume that A < Ao (one proceeds similarly for A > A0). Taking 6 := (1 - Ao)/(l - A) e]0,1[, we have that A0 = dX + (1 - B) • 1 and (1 - A0)*i + A0*2 = 0((1 - A)*i + Xt2) + (1 - 6)t2; so we get the contradiction /((l - X0)h + Xt2) < Of((l - X)h + Xt2) + (1 - 9)f(t2) <6[(l-X)f(t1) + Xf(t2)} + (l-6)f(t2) = (l-X0)f(t1) + Xof(t2). Therefore Eq. (2.9) holds. Taking t€ [h,t2] and A = (t-*i)/(i2-*i) £ [0,1] in Eq. (2.9), we obtain Eq. (2.10). (ii) Let t0 £ dom/ and tx,t2 £ dom/\ {t0}, h < t2. Considering successively the cases t\ < t2 < to, h < t0 < t2, to < t\ < t2, from Eq. (2.16) we obtain that (ft0 is nondecreasing; if / is strictly convex, ipt0 is even increasing. 48 Convex analysis in locally convex spaces (iii) To begin with, let to € int(dom/). Taking into account that ipt0 is nondecreasing on everyone of the nonempty sets dom/n]i0, oo[ and dom/n] — oo,t0[, the following limits exist: ,. ,,x ,. /(*) - /(to) • f /(«) ~ /(to) ^ hm(pto(t) := hm —- = inf -—- < oo, *4-*o t\.to t — to t>to t — to ,. m r fit) ~ /(to) fjt) - f(t0) ^ 1™ V«o(*) := im —:—r— = SUP —:—: > -°°> tjto tfto t — to t If to = max(dom/) then f(t) = oo for t > t0, whence /+(to) = oo; of course f'_ito) < oo (the existence of /i(t0) follows from the monotonicity n of Let now r € [/i(t0),/|(t0)] n E and t < t0 < t'; from Eqs. (2.11) and (2.12) we have that m f - M < f (to) < r < /;(to) < im^lM. / 1 u J 1 t-t0 -• - - ^- - + "'- t'-to Note that the first inequality, in the above relation, is strict if the first quantity is finite and the function / is strictly convex; similarly for the last inequality and the last quantity. From these inequalities we have immedi ately Eq. (2.14), with strict inequality if / is strictly convex. Conversely, assume that r6l and r(t —10) < fit) — /(to) for every t € K; dividing by t — to in each of the cases t > to and t < to and taking the limit for t —> to, we obtain that r 6 [/L(to),/+(to)]- (iv) Let ti,t2 £ dom/, t\ < t2 and consider t €]ti,t2[; using Eqs. (2.11), (2.12) and (2.16) we obtain that r+{h) < /(')-/(«*> < m)-m) = /(«o-/(«a) < , ( } the inequalities being strict if / is strictly convex. Using Eq. (2.13) we get that f'_ and f'+ are nondecreasing, even increasing if / is strictly convex. Suppose that / is not strictly convex; then there exists ti,t2 G dom/, ti < t2, and A0 6]0,1[ such that /((l - A0)ti + A0t2) = (1 - A0)/(ti) + A0/(t2)- Therefore Eq. (2.10) holds, whence fit itl) vte]tut2[-- f'-(t) = fW = ?-{ . t2 — ti Convex functions 49 Because ]ii,i2[C int(dom/), we obtain that f'_ and f'+ are not increasing on int(dom/). (v) Let ti,t2 £ int(dom/), ti < t2 and t,t' e]ti,t2[, t < t'. From Eqs. (2.11), (2.12) and (2.16) we get /;(tlJ+K ); < IMzM < m^M < /('»)-/(«) Vtedom/n]-oo,t0[: /!(*) < f+(t) < /l(*o) €] - oo.oo]. Let A 6 E, A < /i(i0); from the definition of the least upper bound and relation (2.12), there exists *i 6 dom/, ii < to, such that A < (f(h) — f{to))l{t\ - t0). By the left-continuity of / at t0, there exists t2 £]ti,t0[ such that A < (/(ti) - f{h))/{h - t2). Let £ €]*2,*i[- Using again Eq. (2.16) we get A fit) - f(t2) = f(t2) - f{t) J {> t-t2 t2-t - ~ - um = Therefore \im^t0 f'-it) — tt [io,oo[n dom/ if /+(t0) > (>)0. Indeed, if h,t2 E dom/, t0 < h < t2, 0(<) < f'+ito) < f'+ih) < ifih) - fih))Ht2 - h). Similarly, if f_(t0) < (<)0 then / is (decreasing) nonincreasing on ] — oo, to] l~l dom /. So, from the above arguments, if /+(£) > 0 for every t £ int(dom /) then / is nondecreasing on int(dom/); if /+(£) < 0 for every t G int(dom/), by Eq. (2.13), fL(t) < 0 for every t S int(dom/), whence / is nonincreasing on int(dom/). If none of these conditions is satisfied, there exist t\,t2 G int(dom/), h < t2, such that f'+(ti) < 0 < f'+(t2). Let t0 := inf{« € dom/ | f'+(t) > 0} €]h,t2]. It follows that f'_(t) < 0 for every t € dom /, t < t0, whence / is decreasing on ] - oo, t0] C\ dom /. By (v) we have that f'+{to) > 0, whence / is nondecreasing on [t0,oo[ndom/. 50 Convex analysis in locally convex spaces (vii) Let ti, t2 G Ir be such that ti < t2. Then to + A(*i - t0) < h < t2 and t0 + A(tx - t0) < t0 + A(t2 - i0) < t2. So, from the convexity of /, we have x m) < ^ffii^m + \(tl - to)) + t3% l§;Xm), f(to + X(h - to)) < ^ffST^/fa + A(ti - h)) + I-^^L_f(t2), the second inequality being strict if / is strictly convex. Multiplying the first inequality by A, then adding them, we get A/(*i) + /(to + A(t2 - t0)) < /(t0 + A(ti - t0)) + A/(t2), i.e. V'(ti) < ipfo), the inequality being strict if / is strictly convex. The proof is similar on /(. • The preceding theorem shows that /|dom/ is continuous on int(dom/) when / : K -¥ K is a proper lower semicontinuous convex function. We have even more. JS Proposition 2.1.6 Let / G A(R). Then /|ci(dom/) upper semicontinu ous (on cl(dom/)j. Moreover, if f is lower semicontinuous then /|ci(dom/) is continuous. Proof. As mentioned above, / is continuous at any to € int(dom/). Let to G dom/. If dom/ = {to} it is nothing to prove. In the contrary case we can take to = inf(dom/) and some t € dom/ with t > to. Let (t„) C dom/ \ {to} converge to to. We may assume that tn Suppose first that t0 € dom/; then, by Eq. (2.15), /(t„) < =E^/(*o) + V-T/" /(t) for every n. Passing to limit superior we get lim sup/(t„) < /(t0). If to ^ dom/ the preceding inequality is obvious. Hence / is use at to- If / is lsc then /|ci(dom/) is lsc, too. Hence /|ci(dom/) is continuous. • The next theorem furnishes a useful representation of lsc convex func tions on E. This result will motivate the following conventions for a function / G A(K): /l(t) :=/+(t) := -co VfGE\dom/, t Theorem 2.1.7 Let ip : E —> E be a non-decreasing function and a G E 0 fee SMC/I t/m£ V'C ) € E. Then /:E^E, /(*):= J>(s)ds, is a proper lower semicontinuous convex function with I:— {s G R | f'-=V- Proof. As usual, Ja ip(s) ds := — Jb ip(s) ds if a > b. The statement is obvious if / is a singleton. So we assume that inti ^ 0. If t G E \ cl/ it is obvious that f(t) = oo. Hence I C dom/ C cl/. If f(t) = lim^ f(t), and similarly for t = mil. Therefore /|ci/ is continuous, whence / is lower semicontinuous. Let to,ti G /, to < t\ and A G]0,1[. Taking t\ := (1 — A)io + Ati, we have that /(*A) - /(*o) = /£ ¥>(*) ^ < (tA - t0)v(*A) = A(ti - *O)V(*A), f(ti) - f(tx) = Jtl adding them, we obtain that f(t\) < (1 — A)/(to) + A/(*i). Since f\cu is continuous we obtain this inequality also for to,ti G dom/ and A G]0,1[; hence / is convex. Let to G dom/ n cl/ and t 6 E, t > to; because ip(s) > ip+(to) for s > to, we have that f{t) f{to) V(t) > - = -J- /%(s)cfe > and so /'(t0) = ip+(t0). Similarly, fL(to) = ip-{to). Since these relations are obvious for t0 $ dom/ n cl/, Eq. (2.17) holds. Let now g : E —> E be a proper lsc convex function such that g'_ < ip < g'+. It follows that int(domg) = int/. Taking into account Theorem 2.1.5(v) we obtain that g'_(t) = and so g'_(t) = fL{t) and g'+(t) — /+(£) for every t € int I. Taking h : int I —> K, h(t) := f(t) — g(t), we have that h is derivable and h! = 0. It follows that ft is a constant function. Therefore there exists a G E such that g(t) = f(t)+a for all t € int I. Using the preceding proposition it follows that g = f + a. • The following consequence will be used several times. Corollary 2.1.8 Let f : E —• E be a proper convex function. Then \/t,t' e int(dom/) : /(*') - f(t) = //' /;(*) ds = //' /i(s) da. Moreover, if f is lower semicontinuous, then the preceding equalities hold for all t,t' € dom/. Proof. Assume that int(dom/) ^ 0. When t := sup(dom/) e lor t := inf(dom/) € E, we replace, if necessary, f(t) by limt.j.j/(£) and f(t) by lini£4.(/(£); then / is lower semicontinuous. Applying the preceding theorem for ip = f'+ and cp — f'_ the conclusion follows. D The following result is used frequently to show that a function of one variable is convex. Theorem 2.1.9 Let I C E be a nonempty open interval and f : I —> E be a derivable function. The following statements are equivalent: (i) / is convex; (ii) Vt,ae/ : /'(*)•(*-*) 0, i.e. /' is nondecreasing; (iv) ('i// is twice derivable on I) 'it £ I : /"(£) > 0. Proof, (i) =$> (ii) Let / be convex. Since / is derivable we have that f'(s) = /i(s) = /+(s) for every s e I. The conclusion follows from Eq. (2.14) taking r = f'(s). (ii) =$• (iii) Let s,t £ I. By hypothesis we have that /'(*) •(*-*)< /(*) - /(*), /'(*) •(«-*)< /(*) - /(*)• Adding these two inequalities side by side we obtain the conclusion. Convex functions 53 (hi) =4> (i) Let tuh € I, h < t2, A €]0,1[ and tx := (1 - A)fi + Xt2 € }tut2[. Then (1 - X)f(t1) + Xf{t2) - f(tx) = (l-\)[f(t1)-f(tx)] + X(f(t2)-f(tx)) = (1 - AJ/'faXt! - tA) + A/'(r2)(i2 - tx) = A(l - A)/'(n)(ti - i2) + A(l - A)/'(r2)(i2 - tO = A(l-A)(i1-i2)(/'(r1)-/'(r2))>0, with Ti &]ti,t\[, r2 €.]tx,t2[ (obtained by using the Lagrange theorem); of course, we have used the fact that /' is nondecreasing. Suppose now that / is derivable of order 2 on J. In this case (iii) •& (iv) by a known consequence of the Lagrange theorem. • Theorem 2.1.10 Let I C R be an open interval and f : I —> R be a derivable function. The following statements are equivalent: (i) / is strictly convex; (ii) Vt,8el,t?8 : /'(*) •(*-«)< f(t) - f(s); (iii) Vt,s £ I, t ^ s : (/'(*) — f'(s)) • (t — s) > 0, i.e. f is increasing; (iv) (if f is twice derivable on I) Vt E I : /"(*) > 0 and {t e I \ f"(t) — 0} does not contain any nontrivial interval. Proof. The proof is completely similar to that of the preceding theorem; just replace, where necessary, the inequalities by strict inequalities and, of course, use the properties of increasing functions. • Similar characterizations to those of the above theorems can be imme diately formulated for concave and strictly concave functions. As immediate applications of the above two theorems we obtain that: p (i) /i : R -» R, f\(t) := \t\ , where p £]l,oo[, is strictly convex; (ii) /2 : p R+ —> R, /2(£) := t , where p e]0,1[, is strictly concave and increasing; p (iii) /3 : P -> R, f3(t) := t , where p e R!_, is strictly convex and de p creasing; (iv) fi : R -> R, fi(t) :- t if t > 0, /4(t) := 0 if t < 0, where p e]l, oo[, is convex and nondecreasing; (v) /s : R -> R, /s(£) := exp(i), is strictly convex and increasing; (vi) /6 : P -> R, /6(i) := lnt, is strictly con cave and increasing; (vii) f7 : R+ -> R, f7(t) := tint if t > 0, /r(0) := 0, is strictly convex; (viii) f% : R ->• R, f%(t) := Vl +12, is strictly convex. In the following two theorems we characterize the convexity of Gateaux differentiable functions defined on linear normed spaces. 54 Convex analysis in locally convex spaces Theorem 2.1.11 Let D C (X, ||.||) be a nonempty, convex and open set, and /:£)-> E be a Gateaux differentiable function (on D). The following statements are equivalent: (i) / is convex; (ii) \/x,yED : (y-x,Vf(x)) Proof. For x,y G D consider Ix It is well-known that 2 ¥>*,»(*) = V/((l-t)a:+*y)(j/-a:), (i) => (ii) Let x,y € D. By Theorem 2.1.1 we have that (iii) => (i) Let x,y € D and t, s G 7x>y, s < t; then 0<(t-s)( Using Theorem 2.1.9 we obtain that ipX:y is convex, whence, by Theorem 2.1.1, / is convex. Suppose now that / is twice Gateaux differentiable on D. (i) =*> (iv) Let x € D and u € X. Taking into account that D is open, D - x is absorbing; hence there exists a > 0 such that y := x + au G D. From the hypothesis we have that ipx>y is convex, whence by Theorem 2.1.9, < Px,y(t) > 0 for every t G Ix>y. In particular <„(0) = V2/(*)(t/ -x,y-x)=a2 V2f(x)(u,u) > 0. Convex functions 55 Therefore the conclusion holds. (iv) => (i) Let x,y € D. For every t G Ix,y, using Eq. (2.19), we have In the next result we give characterizations for strictly convex functions. Theorem 2.1.12 Let D C (X, \\ • ||) be a nonempty, convex and open set, and f : D -» E be Gateaux differentiable (on D). Then (i) <$• (ii) •£> (hi) •£= (iv), where (i) / is strictly convex; (ii) Vx,yeD,x?y : (y - x, V/(z)) < f{y) - f(x); (iii) Vx,y£D,x^y : (y - x, V/(y) - V/(x)) > 0; (iv) \/x E D, Vw£l\ {0} : S72f(x)(u,u) > 0 (when f is twice Gateaux differentiable on D). Proof. Of course, using Theorem 2.1.1, we know that / is strictly convex if and only if the function A remarkable property of convex functions is given below. Theorem 2.1.13 Let f G A(X) and x G dom/. Then for every u e X there exists f(x, u) := lim ^ + fa)-^} = inf & + tu) ' ™ G I, (2.20) J v ' no t t>o t K ' and VueX : f'(x,u) Proof. Let u £ X and rp : R -> R, tp(t) := f(x+tu). Taking into account that VJ — ifiXtX+u ( 2.1.5 we obtain the existence of ^(Q)=lim^-f^inf^-f), i.e. Eq. (2.20) holds. Using again Theorem 2.1.5 we obtain Eq. (2.21), with the corresponding variant for / strictly convex and u ^ 0. It is obvious that f'{x,0) — 0 and that f'(x,Xu) = Xf'(x,u) for every u G X and A > 0. Let now u,v £ X. We have f(x + t(u + v)) = f (|(x + 2tu) + \{x + 2tv)) < \f(x + 2tu) + \f{x + 2tv), and so f(x + t(u + v)) - f(x) f(x + 2tu)-f(x) f(x + 2tv)-f(x) t ~ 2t 2t ' Letting t ]. 0 we obtain V«,«£l : f'(x,u + v) < f'(x,u) + f'(x,v). Therefore f'(x, •) is sublinear. Note that dom/'(x, •) = E+ • (dom/ - x) = cone(dom/ — a;). If a; € *(dom/) then dom f'(x,-) = lin(dom/ — x), whence 0 € l (domf'(x, •)). From Proposition 2.1.4 we have that f'(x, •) is proper. If a; € (dom /)* then dom f'(x, •) = X and f'(x, •) is proper, which proves that f'(x,u) G M for every u £ X. • The number f'(x, u) G 11 is called the directional derivative of / at x in the direction u. It is possible that f'(x,-) take the value — oo. Consider the function / : R -> E, /(*) := -^1 - i2 if |i| < 1, /(i) := oo if |i| > 1; we have /'(—l,i) = -oo for every t > 0 (Exercise!). The preceding result can be extended in a significant way. Theorem 2.1.14 Let / G A(X), x G dom/ anrf e G M+. Then the function /(a; + to) /(3;)+e fe{x,) :*->!, ^,«):=jnf t- , is sublinear, dom /^(x, •) = cone(dom / - x), (2.22) Convex functions 57 and ft(x,u) f'(x,u) =lim fe{x,u) = inf fe{x,u) Vw £ X. 1 Furthermore, if x £ *(dom/) £/ien f'e{x,-) is proper, while if x £ (dom/) i/ien f'e{x,u) € E /or every u£l. Proof. From the definition of f'e{x, •) it is clear that f'E(x,u) < oo if and only if there exists t > 0 such that x + tu 6 dom f, i.e. u £ cone(dom f — x). Hence Eq. (2.22) holds. It is obvious that f^(x,0) =0 and for A > 0 fs{x,Xu) = inf — A = Xfe(x,u). Let now u,v e X and s,£ > 0. Then / (x + ^(« + «))=/ (^(s + *u) + ^(z + tvj) <^rtf(x + su) + ^rtf(x + tv). It follows that (f(x+£i(u + v))-f(x)+s)/£ +t < f(x + su)- f{x)+e f(x + tv)-f(x)+e ~ s t Therefore, for all s, t > 0 we have /(a! + gu)-/(g)+g , /(a + fa)-/(aQ+e Taking the infimum in the right-hand side, successively with respect to s and t, we obtain that £(x,u + i;) Letting t = 1 in the definition of f'e(x,u) we get Eq. (2.23). On the other hand, observing that for 0 < e\ < £2 < 00 the inequalities VuGX : &(x,u) = f'(x,u) < f'ei(x,u) < f'S2(x,u) 58 Convex analysis in locally convex spaces hold, we obtain x tu x £ ilimf- tu. i.u \ = in• f f'(x,u)tit \ = •m fi- inff f( + )-f(- )+ x tu x = inf inf /(* + *")-/(*)+£ = inf f( + ) ~ f( ) t>0£>0 t t>0 t = f'(x,u) for every u £ X. The other conclusions are proved like in the preceding theorem. • The number f'e{x,u) € K is called the £-directional derivative of / at x in the direction u. We end this section with a characterization of convex functions in ar bitrary topological vector spaces using a generalized directional derivative. We shall give other characterizations in Section 3.2 using generalized sub- differentials. So, let X be a topological vector space and / : X -> M a proper function. We define the upper Dini directional derivative of / at x 6 X in the direction «£lby _1 Df(x u\ _ / limsupt4.0t (/(a;-l-tu)-/(x)) if x € dom/, \ — oo otherwise. When / is convex and x G dom/ from Theorem 2.1.13 we have that Df(x,u) = f'(x,u) for all x € dom/ and u € X. For the proof of the announced characterization we need the following auxiliary result which is interesting in itself, too. Lemma 2.1.15 Let a, b 6 M, a < b, and (p : [a,b] —>• E be a lower semi- continuous proper function with t := sup {Ie]a,b] \ Because ip is lsc, it follows that ip(t) < ip(a) + a(t — a). Assume that t < b. Then, as above, there exists t" E]t,b] such that Df{x,y-x)+Df(y,x-y)<0 for all x, y £ X for which the sum makes sense. Proof. Assume first that / is convex. If x £ dom/ then Df(x,y — x) = —oo, and so Df(x, y — x)-\- Df(y, x — y) equals — oo or does not make sense. Assume that x,y £ dom/. Then Df(x,y- x) = f'(x,y- x) < f(y) - f(x), Df(y, x-y) = f(y, x - y) < f(x) - f(y); summing the inequalities side by side we get the conclusion. We prove the sufficiency by contradiction. Assume that / is not con vex. Then there exists x,y € dom/, x ^ y, and A g]0,1[ such that /((l - X)x + \y) > (1 - A)/(z) + Xf(y). Let I be defined by Df(u, v — u) + Df(v, u - v) 1 1 = s- (p 2.2 Semi-Continuity of Convex Functions In this section X is a separated locally convex space if not stated explicitly otherwise. We begin with some characterizations of lower semicontinuous convex functions. Theorem 2.2.1 Let f : X —> R. The following conditions are equivalent: (i) / is convex and lower semicontinuous; (ii) / is convex and w-lower semicontinuous; (iii) epi / is convex and closed; (iv) epi / is convex and w-closed. Proof. It is well-known that: / is lsc •£> epi/ is closed in I x I « [/ < -M is closed VA £ 1. The equivalence of conditions (i)-(iv) follows immediately applying Theorem 2.1.1. D In the sequel we shall denote by T(X) the class of proper lower semi- continuous convex functions on X. The following criterion for convexity is sometimes useful. Theorem 2.2.2 Let f : X —> K be a proper function satisfying the fol lowing conditions: (i) /(0) = 0, (ii) Vi£l, VA 6 P : /(Ax) = Xf(x), (iii) / is quasi-convex. Suppose that either (a) or (b) holds, where (a) Vi £ X : f(x) > 0, (b) dom/ C cl{a; e X | f(x) < 0} and f is lsc at every point x G dom/. Then f is sublinear. Proof. Note first that f(x + y) < f{x) + f(y) if x,y E dom/ and either f(x) • f{y) > 0 or 0 = f{x) < f(y). Indeed, if f{x) • f{y) > 0 then whence f(x + y) < f(x) + f(y). If 0 = f(x) < f(y), for n £ N we have that nx J£T/(* + V) = f(nTT + n^iV) < max{f(nx),f(y)} = f(x) + f(y). Taking the limit for n —• oo we obtain again that f{x + y) Suppose that (b) holds. Let A := {x G X \ f(x) < 0} C dom/. If x € dom/\^4, by hypothesis, there exists a net (a;i)iej C A, with (xi) —>• a;. Moreover, 0 < f(x) < \iminfieif(xi) < limswpieIf(xi) < 0, whence (/(a;,)) —> /(a;) = 0. Therefore f(x) < 0 for every x € dom/. Let x,y E dom/. Since the roles of x,y are symmetric, we have to consider the following three situations: a) x,y £ A, ft) x,y G dom/ \ A and 7) x G dom/\A and 2/ G A. In the case a) we have that f(x)-f(y) > 0, and so f(x + y) < f(x) + f(y). In the case /3) we have that f(x) = /(y) = 0 and a; + y G dom/, whence f(x + y) < 0 = /(x) + /(y). In the case 7) there exists a net (xi)iei C A such that (xi) —> a;; then (/(a:;)) —>• 0. Since a; + y G dom/ and /(a^ + y) < f(xi) + f(y) for every i e J [by a)], taking the limit inferior we obtain that f(x + y) < f(x) + f(y). • Corollary 2.2.3 Let f : X —> R be a quasi-convex, proper, positively homogeneous and Isc function, /+ := / V 0 and g := f + LC\A, where A := {a; G X I f{x) < 0} U {0}. Then f+ and g are sublinear and f = /+ A g. Proof. It is obvious that /+ and g are quasi-convex, lsc and positively homogeneous. The subhnearity of /+ and g follows applying the preceding theorem. Because f(x) < 0 for x G cl-A, we have also that / — /+ A g. • A result of the same type is given by the following corollary. Corollary 2.2.4 Let f : X —> E be a quasi-convex and positively homo geneous function. If f is upper bounded on a neighborhood of the origin or dimX < 00, then f+ is sublinear. Proof. Suppose that there exists Ao > 0 such that [/ < Ao] is a neigh borhood of 0. Since [/ < A] = ^[/ < A0] for every A > 0, we have 0 G int[/ < A] for every A > 0. Let 0 < A < fi and x G cl[/ < A]. Since 0 G int[/ < A], from Theorem 1.1.2 we have that ^x G int[/ < A]. Hence x G f int[/ < A] = int (f [/ < A]) = int[/ < p] C [/ < »]. Therefore VA > 0 : cl[/ < A] C f| \f Hence [/ < A] is a closed set for every A > 0. Since [/ < 0] = HM>o[^ — lA> the set [/ < 0] is closed, too. Since [/+ < A] = [/ < A] for A > 0 and 62 Convex analysis in locally convex spaces [f+ < A] — 0 for every A < 0, it follows that /+ is lsc. Taking into account that /+ is quasi-convex, /+ is sublinear (applying Theorem 2.2.2). Suppose now that dimX < oo. By hypothesis, [/ < 1] is convex and absorbing. Since dimX < oo, as observed at the end of Section 1.1, we obtain that 0 G int[/ < 1]. The conclusion follows as above. • Note that, under the conditions of the preceding theorem, /+ is contin uous (see Theorems 2.2.9 and 2.2.21). A result similar to that of Proposition 2.1.4 is the following. Proposition 2.2.5 Let f : X —» E be a lsc convex function. If there exists XQ G X such that f(xo) — — oo, then f(x) = —oo for every x G dom /. In particular, if f is sublinear then f is proper. Proof. Suppose that there exists x G dom/ such that f(x) =: t G E. Then (x, t) G epi / and (xo,t — n) G epi / for every n G N. It follows that VnGN : ±(x0,t - n) + (1 - ±){x,t) = (±x0 + *^x,t - l) G epi/; therefore (x,t — 1) G cl(epi/) = epi/, whence the contradiction f(x) < f{x) - 1. • Propositions 2.1.4 and 2.2.5 motivate the consideration, in the sequel, of (almost only) proper convex functions. It is useful to consider the lower semicontinuous envelope or lower semicontinuous regularization / := ci(ePi/) of the function / : X —> E (see Eq. (2.4)). Because cl(epi/) is closed we have that epi/ = cl(epi/). Theorem 2.2.6 Let f : X —> K be a convex function. Then (i) / is convex; (ii) if g : X —> E is convex, lsc and g < f then g < f; (iii) the function f does not take the value —oo if and only if f is bounded from below by a continuous affine function; (iv) if there exists xo G X such that f(xo) = — oo (in particular if f(xo) = — ooj then f(x) = —oo for every x G dom/ D dom/. Proof, (i) Since epi/ = cl(epi/) and epi/ is convex, we have that epi/ is convex; hence / is convex. (ii) Let g be lsc, convex and g < /; then epi/ C epig. Taking the closure of the sets, the conclusion follows. Semi-continuity of convex functions 63 (iii) Suppose that / does not take the value — oo. If f(x) = oo for every x € X then f(x) > (x, 0) + 0 for every x € X. Consider dom / ^ 0 and let x € dom /. Then (x, t) fi epi /, where t := f(x) — 1. Since epi / is a convex, closed and nonempty set, using Theorem 1.1.5 we obtain (x*,a) € X* x E such that V (a;, t) € epi / : {x,x*) + at < (x, x*) + at. Taking x = x and t = f(x) +n, n £ N, we obtain that a < 0. Dividing, by —a > 0, we can suppose that a = — 1. Thus Vz e dom/ D dom/ : f{x)>f(x)>(x,x*)+j, where •y :—t— {x,x*}. Therefore / is bounded from below by a continuous affine function. Conversely, if f(x) > (x,x*) + 7 =: g(x), where x* G X* and 7 £ R, then g is convex and lsc; by (ii) we have that g < f. Therefore / does not take the value —00. (iv) If f(xo) = -00, using the preceding theorem, f(x) — —00 for every x 6 dom/. It is obvious that dom/ D dom/. • Proposition 2.2.7 Assume that f : X —> ffi is sublinear. Then f is sublinear o- / is lsc at 0 •£> / is proper. Proof. Because epi/ = cl(epi/) is a convex cone, the first equivalence follows by Proposition 2.1.2. If / is sublinear, by Proposition 2.2.5 we have that / is proper. Conversely, if / is proper then 0 > /(0) > —00 which implies that 7(0) = 0. • To an arbitrary function / : X —> E one associates, naturally, a lower semicontinuous and convex function; this function is denoted by co/ and has the property that epi(co/) = cl(co(epi /)) and is called the lsc convex hull. It is obvious that co/ < / < /. An application of the lsc convex hull of a function is the property in the next example used in the study of Hamilton-Jacobi equations. Example 2.2.1 Let X be a locally convex space, / € r(X), g : X -> E be a proper function and a, 0 > 0. Then co(co(f+ag)+/3g) =co(f+(a+/3)g). (See the solution of Exercise 2.19 for details.) Related to the upper semi-continuity of a convex function, we remark that when / : X -> E is upper semicontinuous (use for short) at x0 € dom /, 64 Convex analysis in locally convex spaces f is upper bounded (by a real constant) on a neighborhood of xo; therefore xo G int(dom/) 7^ 0. In the sequel we prove that the converse is true for every convex function. Let us first establish the following auxiliary result. Lemma 2.2.8 Let f G A(X) and x0 G dom/. Suppose there exist U G C J4 X and m G ffi_|_ such that VxGx0 + U : f(x) < f(x0) + m. (2.24) Then Vxexo + U : \f(x)-f(x0)\ Proof. Replacing / by g : X ->• R, g(x) = f(x0 + x) — f{x0), we may suppose that xo = 0 and f(xo) = 0. Therefore f(x) < m for every x € U. Prom Proposition 1.1.1 we have that pu is a continuous semi-norm and U = {x G X I pu{x) < 1}. Let x 6 U. Suppose first that t :— pu{x) > 0. Then y :— t~1x € U. Since / is convex and t < 1, we obtain that f(x) < tf{y) + (1 - t)f(0) < tm = mpu{x). Suppose now that pu{x) = 0. Then nx G U for every n € N. It follows that f(x) < ±f(nx) + ^/(O) < \m for every n 6 N. Therefore, once again, f(x) < mpu(x). Since 0 = / (\x' + \{-x')) < \f{x') + y(-x'), we obtain that -f(x') < f(-x') for every x' e X. Using this inequality we obtain that \f(x)\ < mpu(x) for every x eU. Therefore Eq. (2.25) holds. • Using the preceding lemma we get the following important result. Theorem 2.2.9 // the convex function f : X —> K is bounded above on a neighborhood of a point of its domain then f is continuous on the interior of its domain. Moreover, if f is not proper then f is identically -co on int(dom/). Proof. Suppose that there exist Xo £ dom/, V G N(xo) and m G M such that f(x) < m for every x G V. Then V C dom/, and so xo G int(dom /) 7^ 0. If / takes the value -00 then, by Proposition 2.1.4, f(x) = —00 for every x G int(dom/). Therefore the conclusion holds in this case. Let / be proper and x G int(dom/). Then there exists /J, > 0 such that l xi := (1 + fi)x - nx0 G dom /. Taking A := (1 + p,)~ G ]0,1[, we have that / (An + (1 - X)x) < A/(xi) + (1 - \)f(x) < \f(Xl) + (1 - A)m =:mi6R Semi-continuity of convex functions 65 for every x € V. But V\ := Xxi + (1 — A)V £ N(x). Applying the preceding lemma we obtain that / is continuous at x. • Corollary 2.2.10 Let f : X —>• E be a convex function. Then f is continuous on int(dom/) if and only i/int(epi/) is nonempty in X x E. Proof. Suppose that / is continuous at some XQ £ int(dom/). Then, for m &]f(xo),oo[, there exists U £ Nx such that f(x) < m for every x e xo + U. So (xo, m + 1) G int(epi/) because (xo + U) x [m, oo[ C epi/. Conversely, assume that (xo,M £ hit(epi/); then there exist U £ Nx and e > 0 such that (xo + U) x [t0 — e, t0 + e[ C epi /, and so f(x) < m := t0 + e for every x 6 xo + U. By Theorem 2.2.9 we have that / is continuous on int(dom/) (^ 0). • Under the conditions of Lemma 2.2.8 we have a stronger conclusion. Theorem 2.2.11 Let f € A(X) and XQ £ dom/. Suppose that there c exist U € J^ x and m 6 E+ suc/i t/iat condition (2.24) is satisfied. Then for every p e]0,1[, Vx,y £x0+pU : |/(cc) - f(y)\ < m- pu(x-y). 1-p Proof. As in the proof of Lemma 2.2.8, we may assume that xo = 0 and /(a;o) = 0. Let p e]0,1[ and consider x,y 6 pU. We consider two cases: a) pu{x — y) ^ 0 and b) pu{x — y) = 0. In the proof we shall use the fact that pu is a continuous semi-norm, U = {x \ pu{x) < 1} and intf/ = {x | pu{x) < 1} (see Proposition 1.1.1). a) Let tpu{x — y) — pu(—y), and so limt_,.oo ¥>(*) = oo. As (p(l) = pu(x) < P < 1, there exists t > 1 such that ¥>(£) = 1. Then 2 := te+(l-t)j/ £ U. It follows that x = (l-i~1)y + i-1z, and so f(x) < (1 — i~1)f(y) + i~1f(z). From Lemma 2.2.8 we have that \f(y) I < mpu(y) < wp. Therefore /(*) - f(v) < ^ (m - f(y)) < *_1(m + mp). But i{x-y) - z-y, whence ipu(x-y) - pu(z-y) >Pu(z)-pu(y) > 1-p. The preceding inequalities imply that fix) - f(y) < m\^-Pu{x - y). (2.26) 1-p 66 Convex analysis in locally convex spaces b) In this case we have, using again Lemma 2.2.8, that f(t(x — y)) = 0 for every t£i So, WG]0,1[ : f(tx)=f(ty+(l-t)J^(x-y)) Since x G int U C int(dom /), by Theorem 2.2.9, / is continuous at x. From the above inequality we obtain that f(x) < f(y) taking the limit for t —> 1. Hence (2.26) also holds in this case. The conclusion follows from Eq. (2.26) changing x and y. • Corollary 2.2.12 Let (X, ||-||) be a normed space and f G A(X). Suppose that XQ G dom / and for some p > 0 and m > 0, Vz G D(x0,p) : f(x) < f(x0) + m. Then P Vp'e]0,p[,Vx,yeD(xQ,p') : \f(x) - f(y)\ < - • -^ • \\x - y\\. p p-p' Proof. Consider U := D(0;p) — pUx- Then pu{x) = p_1||x|| for any x 6 X. The conclusion is immediate from the preceding theorem. • The conclusion of Theorem 2.2.11 says, in fact, that / is Lipschitz on a neighborhood of xo • We say that the function / : X —> K is Lipschitz on a set A C X if / is finite on A and there exists a continuous semi-norm p on X such that \f(x) — f(y)\ < p(x — y) for all x,y E A; we say that / is locally Lipschitz on A if for every x G A there exists a neighborhood V of a: such that / is Lipschitz on V. Corollary 2.2.13 If f £ A(X) is bounded above on a neighborhood of a point of its domain then f is locally Lipschitz on the interior of its domain. Proof. Let x G int(dom/). Applying Theorem 2.2.9 we obtain that / is continuous at x, and so / is bounded above on a neighborhood of a;. Apply ing now Theorem 2.2.11 we obtain that / is Lipschitz on a neighborhood of a;. • Recall that in Theorem 2.1.5 we have already proved that a function / G A(E) is locally Lipschitz on int (dom/), while in Proposition 2.1.6 we proved that /|dom/ is continuous if / is, moreover, lsc. Another consequence of Theorem 2.2.9 is the following result. Semi-continuity of convex functions 67 Corollary 2.2.14 Let ft 6 A{X) forl 0. Then aff (dom /) is a linear subspace and Y0 := aff (epi /) = aff (dom /) x E. Of course Y0 has finite codimension and is closed. Moreover, by Corol lary 2.2.10, we have that inty0(epi/) 7^ 0. Since epi/ C epig, we have that inty0(y0nepip) ^ 0. It follows (see Exercise 1.3) that rint(epi5) 7^ 0. Since aff (epi g) = aff (dom g) x E, using again Corollary 2.2.10, we obtain that g |aflf(domp) is continuous, and so g is quasi-continuous. • Applying the preceding result to indicator functions one obtains Corollary 2.2.16 Let A C B C X. If A is quasi-continuous then so is B. D There are several classes of convex functions, larger than the class of lower semicontinuous convex functions, which will reveal themselves to be useful in the sequel. We introduce them now. Let / : X —> E. We say that / is cs-convex if n E, Afc/(xfc) .* , ™ fc=l 68 Convex analysis in locally convex spaces x a whenever J2n>i ^n n is convex series with elements of X and sum x € X. Of course, if / is cs-convex then / is convex, while if / is lsc and convex, / is cs-convex (Exercise!). Also, we say that / is ideally convex, bcs- complete, cs-closed, cs-complete, li-convex or lcs-closed if epi / is ideally convex, bcs-complete, cs-closed, cs-complete, li-convex or lcs-closed, respectively. Of course, taking into account the relationships among these notions for sets and Proposition 2.2.17 (i) below, we have: / lsc, convex =>• / sc-convex a- f cs-complete =>• / cs-closed =>• f lcs-closed ^ ^ ^ / bcs-complete =>• f ideally convex => / li-convex => f convex, the reversed implications being not true, in general. Taking into account Proposition 1.2.3 and that for / : X -» R we have [/ < A] = Fix (epi/ n (Xx } - oo, A]), [/< A] = Prx (epi/n (Xx ] - co, A[), the sets [/ < A] and [/ < A] are li-convex (lcs-closed) for every A € R if / is li-convex (lcs-closed). Let A C X; since epi LA = A X R+ and ffi is a Frechet space, we have that A is ideally convex (bcs-complete, cs-closed, cs-complete, li-convex, lcs-closed) if and only if i& is so. Remark 2.2.1 When R is a continuous affine functional (that is R is cs-convex (ideally convex, cs-closed) if and only if / + is so (Exercise!). Proposition 2.2.17 Let f, g : X —>• R have nonempty domains. (i) // / is cs-convex then f is cs-closed. Conversely, if f is cs-closed and is minorized by a continuous affine functional then f is cs-convex. (ii) // / and g are ideally convex (resp. cs-closed) and are minorized by continuous affine functionals then f + g is ideally convex (resp. cs-closed). Moreover, if g is bcs-complete (resp. cs-complete) then f + g is bcs-complete (resp. cs-complete). Proof. Taking into account the Remark 2.2.1 above, when / or/and g is bounded from below by a continuous affine function we may assume that even / > 0 or/and g > 0. Semi-continuity of convex functions 69 (i) The proof of the first part is immediate. Suppose that / is cs-closed and / > 0. Let X)n>i ^nXn be a convergent convex series with elements of X and sum x G X. Since / is convex (epi/ being convex), we may suppose that An > 0 for every n £ N; moreover, we may assume that xn G dom / for every n. Because f(xn) > 0, there exists r := limn-^ X)*!=i ^kfi^k) G EU{oo}. If T < oo, because (xn,f(xn)) € epi/ and J2n>i ^n{xn,f{xn)) = (X,T), and / is cs-closed, we have that (x,r) £ epi/, whence f(x) < r. If T = oo, the preceding inequality is obvious. Hence / is cs-convex. (ii) We prove only the second part of the "ideally convex" case, the rest of the proof being similar. So, let / be ideally convex, g be bcs-complete and /, g > 0. Let J2n>i ^n(xn,rn) be a Cauchy b-convex series with elements of epi(/ + g). Then rn = sn + tn with (xn,sn) G epi/n and (xn,tn) G epig for ev ery n. Because 0 < sn,tn < rn, we have that (sn), (tn) are bounded, s := ^2n>i ^nSn € 1+, t := 52n>1 Xntn G M+ and r = s + t. Because g is bcs-complete we obtain that the b-convex series ^2n>1Xn(xn,tn) with elements of epi g is convergent with sum (x,t) £ epig for some x G X. Hence the b-convex series Yln>i ^n{xn,sn) with elements of epi/ is con vergent with sum (x, s) G epi/. Therefore (x,r) G epi(/ + g). Thus f + g is bcs-complete. • The classes of li-convex and lcs-closed functions have good stability properties. Let us begin with the following characterizations. Proposition 2.2.18 Let / : X -» K have nonempty domain. Then the following statements are equivalent: (i) / is li-convex (resp. lcs-closed); (ii) epis / is li-convex (resp. lcs-closed); (iii) there exist a Frechet space Y and an ideally convex (resp. cs-closed) function F : X x Y -» E such that f(x) = infyCy F(x,y) for every x G X (i.e. f is the marginal function associated to an ideally convex (resp. cs- closed) function). Proof. We prove the "li-convex" case, the proof for the other case being similar. (i) => (ii) Taking % § : X =} E such that gr 31 = epi / and gr S = X x P, we have that "R and S are li-convex multifunctions. Since E is a Frechet space, using Proposition 1.2.5 (iv), epis / — epi/ + {0} x P = gr(IR + §) is li-convex. 70 Convex analysis in locally convex spaces (ii) =$• (i) Since epi,, / is li-convex, as above, the set An := epis / + {0}x ] — i,oo[ is li-convex for every n 6 N. Since epi/ = f]n&NAn, from Proposition 1.2.4 (i) we obtain that epi/ is li-convex. (i) => (iii) Since / is li-convex, there exist a Frechet space Y and an ideally convex set A C Y x X x E such that epi/ = PIXXR(A). Consider the function F: XxYxR-^R defined by F(x,y,r) := r + iA(y,x,r). Then f(x) = inf(a.ir)eepi/r = inf^^^r = inf(j,)r)eyxRF(a;,j/,r) for every x £ X. Since A is ideally convex, so is LA] using Proposition 2.2.17 (ii) and Remark 2.2.1 we obtain that F is ideally convex. The con clusion follows because Y x E is a Frechet space. (iii) =>• (ii) Let f{x) = infyey F(x, y) for every a; € X, where Y is a Frechet space and F is an ideally convex function. From (i)=^(ii) it follows that epis F is li-convex. Then, by Eq. (2.8), epis / is li-convex. • Other useful properties of li-convex and lcs-closed functions are collected in the following result. Proposition 2.2.19 (i) ///„ : X -> E is li-convex (resp. lcs-closed) for every nEN, then supneN /„ is li-convex (resp. lcs-closed). (ii) If fi,h '• X -» E are li-convex (resp. lcs-closed) functions and A € E+, then /i + /2 and A/i are li-convex (resp. lcs-closed). (iii) If F : X xY ^ R is li-convex (resp. lcs-closed) and X is a Frechet space, then h : Y —»• R, h(y) := inixex F(x,y), is li-convex (resp. lcs- closed). (iv) Let Y be a Frechet space and g : Y —> R. If Q C Y is a convex cone, H : X —>• (Y',Q) has li-convex (resp. lcs-closed) epigraph and g is li-convex (resp. lcs-closed) and Q-increasing, then g o H is li-convex (resp. lcs-closed). In particular, if A : X —> Y is a linear operator with li-convex (resp. lcs-closed) graph and g is li-convex (resp. lcs-closed), then g o A is li-convex (resp. lcs-closed). (v) // X is a Frechet space, A : X —> Y is a linear operator with li- convex (resp. lcs-closed) graph and f : X —>• E is li-convex (resp. lcs-closed), then Af is li-convex (resp. lcs-closed). (vi) If X is a Frechet space and /i,/2 : X -> E are li-convex (resp. lcs-closed) functions then /iD/2 and /1O/2 are li-convex (resp. lcs-closed). Semi-continuity of convex functions 71 Proof. Again, we treat only the "li-convex" case. e ne (i) Because epi (supn€N /„) = rineN Pi/™' * conclusion follows using Proposition 1.2.4 (i). (ii) Taking % : X =t R, gr#i := epi/j (i = 1,2), we have that epi(/i + /2) = gr(3?i +3?2)- The conclusion follows from Proposition 1.2.5 (iv). Also, epi(A/i) = T(epi/i) for A > 0, where T:XxE->XxRis the isomorphism of topological vector spaces given by T(x, t) = (x, Xt); hence A/i is li-convex in this case. If A = 0, A/i = tdom/i- But dom/i = Prjt(epi/i), and so dom/i is li-convex by Proposition 1.2.3, whence 0/i is li-convex. (hi) We have, by Eq. (2.8), that epis / = PryXR(epis.F). Since episF is li-convex and X is a Frechet space, we get from Proposition 1.2.3 that epis / is li-convex. (iv)-(vi) Using the same constructions as in the proofs of Theorem 2.1.3 (vi), (viii) and (ix), the conclusions follows from (iii). • If X is a barreled space, the lower semi-continuity of a convex function (even weaker conditions) ensures its continuity on the interior of its domain. Theorem 2.2.20 Let X be a barreled space and f : X —> E be convex. Suppose that either (a) X is first countable and f is li-convex or (b) / is lower semicontinuous. Then (dom/)' = int(dom/) and f is continuous on int(dom/). Proof, (a) By Proposition 2.2.18, there exist a Frechet space Y and a li-convex function F : X xY -> R such that f(x) = infy€y F(x, y) for every x £ X. Consider the multifunction X:7 xlrj I with gr3? := {(y,t,x) \ (x,y,t) G epiF}. Of course, Ji is ideally convex and ImR = dom/. Let 1 _1 xo € (dom f) (if this set is nonempty) and {ya,tQ) € 3l (a;o). Applying Simons' theorem (Theorem 1.3.5), we have that U := R(Yx ] - 00, t0 + 1[) is a neighborhood of x0. So, for every x € U there exist y 6 Y and t < to + 1 =: m such that f(x) < F(x,y) Proof. By Proposition 1.2.1 we have that epi/ is cs-closed, and so / is lcs-closed. The conclusion follows from the assertion (a) of the preceding theorem. • The application of Corollary 2.2.12 and Theorem 2.2.20 yields the fol lowing uniform boundedness principle for convex functions. Theorem 2.2.22 Let X be a Banach space and C C X be an open convex set. Consider {fi)i^i o, nonempty family of continuous convex functions from C into E. If (fi{x)).j is bounded for every x £ C then for every x £ C there exist rx,Lx > 0 such that Ux := x + rxllx C C and \fi(y) — fi(z)\ < Aclly - z\\ for all y,z £ Ux and all i £ I (i.e. (fi) is locally equi-Lipschitz on C). Proof. By hypothesis, for every x £ C there exists Mx £ E+ such that \fi(x)\ < Mx for all i £ J. Let r'x > 0 be such that U'x := x + r'xUx C C and consider Fi : X -> E be denned by Fi{y) := fi(y) for y £ U'x and Fi(y) := oo otherwise. Then Ft £ T(X). Consider F := supieI Ft E T(X). The hypothesis shows that domF = Ux. By Theorem 2.2.20 we obtain that F is continuous on int(dom.F) — x + rxBx- Therefore there exist r 'x £]°><[ and M > 0 such that F(y) < M for all y G x + r'J,Ux. Then for y G x + r'J.Ux and i G / we obtain that ft(y) - fi(x) < M — (-Mx) = M+Mx =: L'x. Using now Corollary 2.2.12 we have that fi is Lipschitz with constant Lx :~ ZL'x/r'^ oiix + rxUx, where rx := r'^/2. The conclusion follows. • Taking X a Banach space, Y a normed space, and {Xi | i G 1} C £(X, Y) (I 7^ 0) with {TiX \ i £ 1} bounded for every x £ X, the conditions of the preceding theorem are satisfied by the family of functions (fi)iei, where U{x) := ||Ti(a:)||, and C := X. Hence \\Ti(x)\\ = \h(x) - /4(0)| < L||x|| for x £ rllx and i £ I, for some r,L > 0; hence \\Ti\\ < L for every i £ I. So we obtained the classic uniform boundedness principle in Functional Analysis. From the preceding result we obtain that pointwise convergence implies uniform convergence on compact sets. Corollary 2.2.23 Let X be a Banach space and C C X be an open convex set. Consider (/„) a sequence of continuous convex functions defined on C with values in E such that (fn(x)) -> f(x) for every x £ C with f : C —> E. Then (/„) converges uniformly to f on the compact subsets ofC Semi-continuity of convex functions 73 (or, equivalently, (fn{xn)) -> f(x) for every sequence (xn) C C converging to x E C). Proof. First of all observe that / is convex (by Theorem 2.1.3) and locally Lipschitz (by the preceding theorem). Let x,xn E C (n E N) be such that (x„) -> x. By Theorem 2.2.22 there exist r, L > 0 such that Ux := x + rUx C C and \fn{y) - fn(z)\ |/n(*n) - f(x)\ < \fn(xn) - fn(x)\ + \fn(x) - f(x)\ Since K is compact, we may assume that (xn) -> x 6 K...C C. Then, as shown above, {fn{xn))p -> f{x). Since / is continuous, we have also that (f(xn)) p —>• f(x), which yields the contradiction 0 > e. • The next result corresponds to a known theorem for continuous linear operators. Proposition 2.2.24 Let X be a Banach space and C C X be an open convex set. Assume that /, /„ : C —> M. (n € N) are continuous convex x functions. Then (fn( )) —> f(x) for every x 6 C if and only if (a) (fn{x)) is bounded for every x G C and (b) (fn(x)) -> f{x) for every x € D for some dense subset D of C. Proof. The necessity is obvious. Assume that conditions (a) and (b) above are satisfied but (fn{x)) does not converge to f(x) for some x E C. Hence there exist e > 0 and P C N an infinite subset such that \fn(x) - f(x)\ > e for every n E P. By Theorem 2.2.22 we find r,L > 0 such that Ux := x + rUx C U and /,/„ (n E N) are L-Lipschitz on Ux. Since D is dense, there exists (xk) C D converging to x; we assume that Xk E Ux for every fcgl Then for n E P and k E N we have S < \fn(x) - f(x)\ < \fn(x) - fn(Xk)\ + \fn(xk) ~ f(xk)\ + \f(xk) - f(x)\ < 2L \\xk - x\\ + \fn{xk) - f{xk)\. 74 Convex analysis in locally convex spaces Fixing k G N such that \\xk - x\\ < e/(4L), we get |/„(xfc) - f(xk)\ > e/2 for n G P, contradicting that (fn(xk) - n G N) converges to f(xk). • Let / G T(X); we call the recession function of / the function /oo : X —>• IK whose epigraph is (epi/)^. Let XQ G dom/; taking into account formula (1.3), we have that (u, A) G epi/oo <»Vi> 0 : (x0,/(x0)) + £(u, A) G epi/ /(x0 + fa) - /(x0) < ^ Jim /(X0 + fa) - /(x0) = gup /(Xp + fa) - /(XQ) < A t-S-OO £ (>0 t ~ Therefore VtiSl : /oo(u) = lim 7 ; (2.27) t—>oo r thus /oo(u) > -oo for every u G X and /oo(0) = 0. Because epi/oo is a closed convex cone we have that f^ is a lsc sublinear functional. The preceding relations show that f(x + u) < /(x) + /oo(u) VxGdom/, Vu G X, (2.28) [/ < A]^ = {u e X | /«,(«) < 0} VAGEwith[/ Remark 2.2.2 Note that, for / G F(X) and x G X, the mapping t >->• /(x + fa) is nonincreasing from E into E when /oo(u) < 0; in particular dom/ + E+u = dom/. Moreover, if /<»(«) < 0 and /oo(~w) < 0 then f(x + tu) = f{x) for all x G X and t G E. Indeed, let ti < t2. If /(x + t\u) < oo then /(x + t2u) - f(x + tiu) _ f(x + tin + (t2 - h)u) - f(x + tiu) t2 —t\ t2 — *i Conjugate functions 75 and so f(x + t2u) < f(x + t\u); the inequality is obvious if f(x + tiu) = oo. It follows that dom/ + R+u = dom/. When /«>(«) < 0 and /oo(-w) < 0 we can apply the previous result for u and — u and obtain that the mapping t H-> f(x + tu) is constant. 2.3 Conjugate Functions In this section X and Y are separated locally convex spaces if it is not stated explicitly otherwise. Let / : X —> M; the function f*:Xm-+% f*{x*):=suv{(x,x*)-f(x)\xeX}, (2.30) is called the conjugate or Fenchel conjugate of /. Note that if there exists XQ 6 X such that /(xo) = ~°o then f*(x*) — oo for every x* € X*, while if f(x) = oo for every x then f*{x*) = -oo for every a;* £ X*. When / is proper (but also for / not proper, using the convention inf 0 = oo), we have f*(x*) = sup{(x,a;*) - f(x) | x € dom/}. The conjugate of a function h : X* -> K. is defined similarly: /i* :X->K, /i*(a;) := sup{(a;,a;*) - h{x*) \ x* e X*}. In fact, considering the natural duality between X and X*, it is the same definition; this distinction is useful in the case of normed spaces. The above remark concerning /* is also valid for h*. In the following theorem we establish several simple properties of con jugate functions. Theorem 2.3.1 Let f,g : X -> I, h : Y ->• 1, k : X* -)• I and A€l(X,Y). (i) /* is convex and w*-lsc, k* is convex and Isc; (ii) the Young-Fenchel inequality below holds: Vxel.Vi'er : f{x) + f*(x*)>(x,x*); (iii) f<9^9* (v) if a > 0 then (af)*(x*) = af'ia^x*) for every x* G X*; if/3 ^ 0 tfien (f(l3-))*(x*) = f'ifi-^x*) for every x* G X*; (vi) if g(x) = f(x + x0) for x G X, then g*(x*) = f*(x*) - {x0,x*) for every x* G X*; (vii) if x*0 G X* then (/ + x*0)*{x*) = f*(x* - x*0) for every x* G X*; (viii) if f, h are proper and $ : I x 7 -> 1, $(x, y) := f(x) + h(y), then $*(x*,y*) = f*(x*) + h*{y*) for all (i*,y*) E X* x Y*; (ix) {AfY =f*oA* and (fDg)* = f* + g*. Proof, (i) If / is not proper, we have already seen that /* is constant, and so /* is convex and w*-continuous. If / is proper we have that /* = su Pzedom/ *Px, where ipx : X* -*• R, ipx(x*) := {x,x*) - f(x). It is obvious that for every x G dom/, (px is affine (hence convex!) and u>*-continuous (hence w*-lsc!). Therefore /* is convex and u;*-lsc. For the statement about h we use the same arguments. (ii) By Eq. (2.30) we have VxeX,Vx* eX* : /'(a;*)> (Afy(y*) = sup{(y,y*) - (Af)(y)] = sup ((j,,**) - inf f(x)) y£Y y£Y V {x\Ax=y} J = sup{(y,y*) - f{x) \(x,y)£X x Y, Ax = y} = sup{(x,A*y*)-f(x)\x£X} = f*(A*y*) = (roA*)(y*). Similarly one proves the corresponding relation for (fDg)*. D Conjugate functions 77 I the sequel we denote by T*(X*) the class of those functions in A(X*) which are w*-lower semicontinuous. The discussion at the beginning of this section and the preceding theorem yields the next result. Corollary 2.3.2 Let f : X -> E and h : X* -• E. Then (i) /* € T*(X*) Theorem 2.3.3 (of the biconjugate) Let f e T(X). Then f* € T*{X*) and f **:=(/*)* = f. Proof. Applying Theorem 2.2.6 we get x$ G X* and a G ffi such that Viel : /(i) > (x,x*0)+a. (2.31) Thus, by the preceding corollary, we have that /* G T*(X*). Moreover, Theorem 2.3.1 shows that /** < /. Let x € X be fixed and consider t G E such that t < f(x); therefore (x, t) ^ epi/. Applying Theorem 1.1.5 for {(x,t)} and epi/, there exist (x*,a) G X* x E and A G E such that V(z,i)Gepi/ : (x,x*) + ta < A < (x,x*) + ta. (2.32) Taking (x, t) = (x, f(x) + n), n G N, with x G dom/, we obtain that VnGN : (x,x*) + af(x) + na < A < (x,x*) + ta. Letting n -> oo we obtain that a < 0. Take first a < 0. Dividing eventually by —a > 0, in Eq. (2.32) we can suppose that a = —1. Thus (x,x*) - f(x) < A Vi£ dom/. Hence /*(#*) < A < (x,af) - <, and so t<(x,x*)-r(r) VzGdom/ : {x,a;*) + c < (x, x*). 78 Convex analysis in locally convex spaces This together with Eq. (2.31) yields f(x) > (x, XQ) + a > (x, XQ) + a + t(x, x*) + tc — t(x, x*) for all x € dom / and all t > 0; this implies successively: MxeX, Vi>0 : -tc + t(x,x*) - a> (x,XQ+tx*) - f(x), V£>0 : -tc + t{x,x*) -a > f*(x*Q + tx*), Vi>0 : f**(x) > (x,x*0+tx*)-f*(x*0+tx*) > a+ tc+(x,x*0). But there exists t > 0 such that a + tc + (x, a;*,) > t; hence f**(x) > t in this case, too. Thus we obtained f**(x) > f{x). Therefore /** = /. • The preceding theorem shows that for any function / : I -> 1 we have ((/*)*) = /* (for the duality (X,X')) which shows that there is no interest to consider conjugates of order greater than 2. It also shows that the conjugation is an isomorphism between T(X) and T*(X*). More precise information on the biconjugate of an arbitrary function is furnished by the following result. Theorem 2.3.4 Let f : X —>• ffi have nonempty domain. (i) // co/ is proper, then /** = co/; if co/ is not proper, then /** = —oo. (ii) Suppose that f is convex. If f is Isc atxG dom/, then f(x) = /**(x); moreover, if f(x) € R, then /** = / and f is proper. Proof, (i) The function co/ is convex and lsc. If co/ is proper, using the preceding theorem and Theorem 2.3.1(iv), we have that co/= (co/r = (/•)* = /**• If co/ is not proper, since dom(co/) D dom / / 0, co/ takes the value -co; hence /* = (co/)* = oo, and so /** = —oo. (ii) Taking into account that / is convex, co/ = /. Since / is lsc at x, we have that f(x) = f(x). It is obvious that f**(x) = f(x) if f(x) = -oo. Let f(x) £ M; then f(x) € E, and so / is proper. From the first part we have that /** = J, whence f**(x) = J(x) = f(x). • Corollary 2.3.5 Let f,g £ A(X). If fUg is proper, then (/* + g*)* = fOg = fOg~, while in the contrary case (/* + Vx G int(dom(/D<7)) = int(dom/) + domg : (fDg)(x) = (f*+g*)*(x). The sub differential of a convex function 79 Proof. From Theorem 2.3.1 we have that uogr = r+g* = T+r = (f^9)*\ since /• sA:X*^W, sA(x*) :=sup{(x,x*)\x &A}; (2.33) for 0 ^ B C X* the support function SB : X —> E is defined similarly. It is obvious that SA is w*-lsc and sublinear while SB is lsc and sublinear. Furthermore, if C C X is another nonempty set then SA+C — $A + $c and SAUC — SA V sc- Note that (IA)*(X*) = sup(a;,a;*) = sA(x*) = sup (x,x*) = (LCOA)*(X*) = ScoA- x£A X€COJ4 Moreover we have that icoA — cotyi and dom SA = dom(tJ4)* = {x* € X* | x* is upper bounded on A}; therefore doms^ = X* if and only if A is u;-bounded. In the next section we shall see that LA is useful in determining the normal cone of C. 2.4 The Subdifferential of a Convex Function We have already seen in Section 2.1 that if the proper convex function / : (X, || • ||) -> M. is Gateaux differentiable atiG int(dom /) then Vzedom/(Va;eX) : (x -x, Vf(x)) < f(x) - f(x). Taking into account this relation, it is quite natural to consider the elements x* E X* which satisfy the inequality VxEX : (x-x,x*) In this section the spaces under consideration are separated locally con vex spaces if not stated otherwise. Let / : X -¥ E and x £ X be such that f(x) G E. An element x* G X* is called a subgradient of the function / at x if relation (2.34) is satisfied; the set of all the subgradients of the function / at x is denoted by df(x) and is called the subdifferential or Fenchel sub differential of / at x. We consider that df (x) = 0 if f(x~) £ E; of course we can have df(x) = 0 even if f(x) G E. Thus we obtain a multifunction df : X =4 X*. By the preceding considerations we have that domdf C dom/. We say that / is subdifferentiable at x G X if df(x) ^ 0. Note that if x* G df(x), the afflne function tp : X —• R, (p(x) := (x,x*) — (x,x*) + f(x) minorizes / and coincides with / at x; this proves that V(x,i)€epi/ : (x,x*) — t < a := (x,x") — f(x), which shows that the hyperplane {(x,t) € X x R \ (x, x*) — t • 1 = a] is a, non vertical (since the coefficient of t is ^ 0) supporting hyperplane (see p. 5 for the definition). Recall that in Theorem 2.1.5 we have already determined the sub- differential of a function / 6 A(M) at to £ dom /: 0/(*o) = [/:(*o),/;(to)]nR. In the sequel we shall establish properties of the subdifferential and methods for computing it also for X ^ E. In the next result we collect several easy properties of the subdifferential. Theorem 2.4.1 Let f : X -> 1 and x 6 X be such that f(x) 6 E. Then: (i) df(x) C X* is a convex and w*-closed (eventually empty) set. (ii) Ifdf(x)^ (co/)(x) = J(i) = f(x) and d(cof)(x) = dj(x) = df(x); in particular /** = co/ and f is proper and Isc at ~x. (iii) // / is proper, dom f is a convex set and f is subdifferentiable at every x £ dom /, then f is convex. Proof, (i) Let x\, x*2 e df{x) and A G ]0,1[. Then VzGX : (x-x,x*1) Multiplying the first relation with A > 0, the second with 1 - A > 0, then adding them, we obtain Vx£l : (x-x,Xx*1+(l-X)x*2) /(xo) - fix). Let a G E be such that {x0 — x,x*) > a > f(x0) — f{x). Then V := {x* \ (x0 -x,x*) > a} is a neighborhood of x* for the topology w* = a(X*,X). It is obvious that V n df(x) - 0. So df(x) is w*-closed. (ii) We already know that co/ < f < f- Let x* G df(x) and y>:K->K, ip(x):=(x-x,x*) + f(x); (fi is convex, continuous and Vx£l : (x-x,x*) < (cof){x) - (co/)(z) < f{x) - f(x), whence d(cof)(x) C df(x). Therefore d(cof)(x) = df(x) = df(x). (iii) By (ii) we have that f(x) = cof(x) for every x G dom/. Since co/ is a convex function and dom / is convex, it is obvious that / is convex. • The property (ii) of Theorem 2.4.1 justifies why we consider only proper convex functions when discussing subdifferentiability (in the sense of convex analysis!). In a similar way we introduce the subdifferential of a function h : X* —> lata point x* G X* with h{x*) G E: dh(x") = {x € X | Vx* G X* : {x,x* - x*} < h(x*) - h(x*)}. (As for conjugation, this distinction is important only when working with normed spaces; in the case of locally convex spaces we always consider the natural duality between X and X*, when the dual of X* is X.) Similar results to those of Theorem 2.4.1 hold; more precisely dh(x*) is a closed convex (eventually empty) set and if dh(x*) ^ 0 then h is proper and w*-\sc at x*; the other statements are also valid if one takes the closures with respect to the weak* topology. 82 Convex analysis in locally convex spaces In practice (for example for solving numerically problems using com puters) it is possible to determine the subgradients only approximately. In this sense the following notion of subgradient reveals itself to be useful. Let / : X ->• 1, x G X with f(x) G M. and e G !+; the element x* G X* is called an e-subgradient, of the function / at x~ if Vxel : (x-x,x*) Of(x) = d0f(x) C 3ei/(af) C 5ea/(i). Moreover VeGl^ : 9e/(i) = fl a,/(i). The £-subdifferential of a function /t: I* -> K at ? € I* with /i(x*) G E is introduced similarly. In the following theorem we collect several properties of the subdifferen tial and of the £-subdifferential. Before stating this theorem, let us intro duce some notions: we say that the set M C X x X* is monotone if V(x,x*),(y,y*)€M : (a: - y,x* - y*) > 0; M is strictly monotone if V{x,x*), (y,y*)eM, x^y : (x - y,x* - y*) > 0; M is maximal monotone if a) M is monotone and b) M' C X x X* monotone and M C M' imply M = M', i.e. if M is a maximal element of the class of monotone subsets ordered by inclusion. We say that the multifunction T : X =3 X* is monotone, strictly monotone or maximal monotone if gr T is monotone, strictly monotone or maximal monotone, respectively. Of course, when X = E and T is single-valued (i.e. T(a;) is a singleton for every x G domT), T is (strictly) monotone if and only if the function T|domT is nondecreasing (increasing). In the next result we do not assume that the functions are convex. The subdifferential of a convex function 83 Theorem 2.4.2 Let f,g : X ->• R, h : Y -> R be proper functions, A 6 L(X,Y), x G domf n domg, y G dom/i and e E M+. T/»en: (i) dsf(x) is a convex w* -closed set. (ii) x* G 3e/(z) «• /(x) + f*(x*) <(x,x*)+e^x£ def*(x*). (iii) a;* G df(af) «• /(x) + /*(a;*) < (x,x*) & f(x) + f*(x*) = (x,x*) => xedf*(x*). (iv) dom(<9£/) C dom/, lm(dsf) C dom/* and df is monotone. (v) 3/(3;) ± 0 «• /(3f) = maxa.ex. ((x,x*) - f*{x*)). (vi) de(f + x*)(x) = x* + def(x) for x* E X*; 0e(A/)(3:) = \ds/xf(x) and d(Xf)(x) — Xdf(x) for X > 0; if g(x) = f(x + XQ) for x G X then dEg(x) = dEf(x + x0). (vii) Assume that y = Ax; then A* (deh(y)) C de(h o A)(x). vii d ( i) U^6[0,e] ( vf(*) + 9S-Vg(x)) C3£(/ + g)(x). (ix) Suppose X is a normed space, f is Isc, £j > 0 and (xi,x*) G grdEif for every t £ 7. If (ei)ig/ —> e < oo and either (a) (xi)iei —• a;, (x*)iei -^-> a;* and (x*)ie/ is norm-bounded or (b) (x»)ie/ -^-> £, (a;j)jej is norm-bounded and (x*) —t x*, then (x,x*) G grdsf. In particular grdef is closed in X x X* (for the norm topology). Proof, (i) The fact that dEf(x) is convex and u>*-closed is shown similarly to the first part of Theorem 2.4.1. (ii) We have that x* ed£f(x) oVxGX : (x - x,x*) < f{x) - f(x) +e »Vi£l : {x,x*)-f(x)<(x,x*)-f(x) + e &f*(x*)<(x,x*)-f{x)+e &f{x) + f*(x*)<(x,x*)+e. Assume now that x* G def(x). Then, by what precedes, f"(x) + /*(**) < f(x) + fix*) < (x,x*)+e, and so x € d£f*(x*). (iii) We already know that (37, x*) < f(x) + f*(x*) (the Young-Fenchel inequality!); the equivalences follow now from (ii) taking e = 0. (iv) It is obvious that dom(<9£/) C dom/, while the inclusion Imdef C dom/* follows from (ii). 84 Convex analysis in locally convex spaces Let (x,x*), (y,y*) £ gr<9/. Then x,y £ dom/ and {y - x, x*) < f(y) - f(x), (x - y,y*) < f(x) - f(y). Adding these relations we get (y — x,x* — y*) < 0, and so df is monotone. (v) If x* € df(x), taking into account (iii) and the Young-Fenchel inequality, we have Vx'el* : (x,x*)-r(x*)=f(x)>(x,x*)-r(x*), and so the implication "=^" is true. The converse implication is an imme diate consequence of the equivalences of (iii). (vi)-(viii) follow easily from the definition of the e-subdifferential. (ix) Suppose that X is a normed space. Let (e'j)ie/ -> e < oo and ((xi,x*))i€l have the properties from (a) or (b). Taking into account that \X{, 3Jj j \Xy X / — \X{ Xj X^ J ~\- \X) X* -X*) = (Xi,X* -X*) + (Xi-X,X*) for every i, we get ((y — Xi,x*)) —t (y — x,x*) for every y 6 X in both cases. But Vie/, Vy£X : f(xi) + (y-xi,x*) Y" (xi+1-Xi,x*)<0, (2.36) *—'i=0 where xn+i := XQ. Taking n = 1 in Eq. (2.36) we obtain that {x\ - x$, x0') + (x0 — xi,x\) < 0, i.e. (xi —xQ,xl — XQ) > 0, for all (X0,XQ), (XI,XI) £ grT. Hence every cyclically monotone multifunction is monotone. When / : X -> R is a proper function, df is a cyclically monotone multifunction. Indeed, let n e N and ((xi,a;*))"=0 C gr<9/; then (xi)™=0 C dom/ and (xi+i -xt,x*) < f(xi+i) - f(xi) Vi, 0 < i < n, where, as above, xn+\ := XQ. Adding these relations side by side for i = 0,1,..., n, we get Eq. (2.36). Note that to a nonempty cyclically monotone multifunction one can associate a function / 6 T(X). The subdifferential of a convex function 85 Proposition 2.4.3 Let T : X =4 X* be a cyclically monotone multi function and (icc^o) £ gr^- Consider fr : X —>• M defined by a: 1 2 37 h{x) := sup f (a; - zn, <) + X) .=Q ( *+ ~ ^ ^) ) > ( - ) where the supremum is taken for all families ((x,,x* ))"=0 CgrT withne N. T/ien /T e T(X), /T(XO) = 0 and T(x) C dfT(x) for every x eX. Proof. Because fr is a supremum of continuous afHne functions, fa is lsc, convex and nowhere -oo. Note that domT C dom/y. Indeed, let (x,x*) G grT. Taking k G N, ((a:i,xj))*=0 C grT, n := fc + 1 and (x„,<) := (i,?), from Eq. (2.36) we obtain that Eit-i _ . {Xi+i - Xi, X*) + (X- Xk,X*k) + (X0 - X, X*) < 0, 1=0 whence fr{x) < (x — x0,x*). Hence x G dom/y. In particular the preced ing inequality shows that /T(^O) < 0. Taking n = 1 and (xi,x*) = (XO,XQ) in the definition of fr(xo), we get fr(xo) > 0. Hence /T(^O) = 0. Therefore h € T(X). Let now (x,x*) G grT and x G X. Consider an arbitrary it < fr{x) + (x — x,x*). Then there exist k G N and ((XJ,X*)J C grT such that _ r—\fc-i /x- (x-x,x*) < {x-xk,x*k) + 2^.=0 fci+i -a;»,a;*>- Taking n = k + 1 and (xn,a;*) := (x,x*), the preceding inequality yields n—1 Ei=o (Xi+1 ~XuX^ - -^W- Letting (i —¥ fr (x) + (x — x, x*), we obtain that fr (x) + (x — x, x*) < fa (x), and so x* G 9/r(S). D In the next theorem we use the convexity of the function /. Theorem 2.4.4 Let f G A(X), x G dom/ and e G ffi+ . T/»en: 1 (i) def(x) = df'e(x, -)(0); moreover, ifx£ (dom/) and / is Gateaux differentiate at x then df(x) = {V/(x)}. (ii) 7/ / is strictly convex then df is strictly monotone. (hi) / is lsc atx if and only ifd£f(x) ^ 0 for every e > 0; d£f*(x*) ^ 0 if f (x*) eRande>0. (iv) Suppose that f is lsc at x. Then x* G dEf(x) <£> x G def*(x*). 86 Convex analysis in locally convex spaces Proof. (i) Ii x* £ dsf(x), replacing x by x + tu, with t > 0, in Eq. (2.35), then dividing by t we obtain V«€l : (u, x ) < inf — ^-^ — fe{x,u), and so a;* £ df^(x, -)(0). The converse inclusion follows from the inequality f'e(x, u) < f(x + u) — f(x) + e, valid for every u £ X. If / is Gateaux differentiable at x then f'+{x, •) — V/(aT); the conclusion is obvious. (ii) Let x,y £ dom/, x / y, and x* € df(x), y* £ df(y). From (i) we have that (y-x,x*) < f+(x,y-x) < f(y)-f(x), & - y, y*) < f+(y, x-y)< f(x) - f(y), because / is strictly convex. Adding the above two relations we obtain that (y — x, x* — y*) < 0. This shows that df is strictly monotone. (iii) Suppose that / is lsc at x and let us take e > 0; using Theorem 2.3.4 we have f(x) = r(x) = sup{(z,z*) - f*(x*) I x* G X*} > f(x)-e. Therefore there exists a;* £ X* such that {x, x*)—f*(x*) > f(x)—s, whence x* £ def(x). Conversely, suppose that dEf(x) ^ % for every e > 0. Then Ve>0, 3x*£X* : f(x) - e < {x,x*} - f*(x*) < f(x), whence f(x) < f**(x). Since /** 0. By the definition of /*, there exists x £ X such that f*{x*) < (x,x*) — f(x) + e, whence (x,x*-x*) < (x,x*) - f(x) - f*(x*)+e< f(x*) - f*(x*)+e, i.e.x£def*(x*). (iv) Since / is lsc at x, we have that f**(x) = f{x); the conclusion is immediate using Theorem 2.4.2 (ii). • Remark 2.4-1 Note that for / 6 A(X), x £ dom/ and U £ Mx{x) we have that df(x) = d(f + iu)(x), but def{x) and d£(f + iu)(x) may be The subdifferential of a convex function 87 different for e > 0, i.e. df is a local notion while dsf (with e > 0) is a global one for convex functions. Indeed, since f'(x, u) — (f + t[/)' (x, u) for every u £ X, the first remark follows from assertion (i) of the preceding theorem. For the second remark let us consider / € A(R) given by f(x) = x2 and U = [—1,1]. Then ds{f + iu)(0) = def{0) = [-2y/i,2y/e\ for e € [0,1] and de{f + ta)(0) = [-1 -£,l + e]£ [-2y/i,2^/i\ = 0e/(O) for e > 1. Generally, the formula d(\f)(x) = Xdf(x) is not true for A = 0; this formula, however, is true if x £ (dom/)1. In assertions (vii) and (viii) of Theorem 2.4.2 we have equality only under supplementary conditions called generally "constraint qualification conditions;" in Section 2.8 we shall give such conditions. Let A C X be a nonempty convex set and a e A; then diA{a) = {x* eX*\Vx£A : (x-a,x*)<0} = {x* |VxG A : (x,x*) < (a,x*)} = {x* | Vx 6 cone(A - a) : (x,x*)<0} = —(cone(A — a))+ = —cone(A — a)+. The set <9M(O) is denoted by N(A;a) and is called the normal cone of A at a S A, while the set cone (A — a) is denoted by C(A;a) (even if A is not convex); evidently, the set C(A;a) is a closed (convex if A is so) cone. Taking into account the relation established above and the bipolar theorem (Theorem 1.1.9), we have that N{A;a) = -(C(A;a))+ and C(A;a) = -(N(A;a)) + . It is obvious that N(A; a) = {0} if a £ A*. We observe that x* € N(A; a) \ {0} if and only if Hx.^a^x*) is a supporting hyperplane to A at a and Ac//-,, ,,. The set df(x) may be empty even if / is lsc at x. For example, the function / : R -> E, f(x) := -Vl - x2 for \x\ < 1, f(x) := oo for |a;| > 1 (already considered in Section 2.1) is lsc, finite at 1, but df(l) = 0. Moreover 9(0-/) (1) = R_. In the preceding section we have met some situations in which it was easy to compute the conjugate of a function. We shall point such situations for the e-subdifferential, too. 88 Convex analysis in locally convex spaces Corollary 2.4.5 Let fi : Xi —> R for i E \,n be proper functions and x = (xi,... ,xn) E Yli=i dom/j. Let us consider the function nn — • ^n . Xi-tR, 9 £i £l H de /n particular nn . dfi(xi). Proof. We have already seen in the preceding section (for n = 2, but the x extension is immediate) that y*{x\,... ,xn) = Y12=ifi( i)- Therefore, by Theorem 2.4.2 (ii), x* E de O 3 e\,..., en > 0, ei + ... + en = e, Vi G T~n : /i(xi) + /*(z*) - (xi,x*) < et, O 3 £i, . . . , 6n > 0, £! + ... + £„ = £, Vi G l,n : x* € dEifi(xi), whence the conclusion. • Corollary 2.4.6 Let f : X -» R be a proper function and A E &(X, Y). If x E dom/ and y E Y are such that y = Ax and (Af)(y) = f(x), then for every e E R+ we have 1 de(Af)(y)=A'- (def(x)). Proof. Using Theorem 2.4.2 (ii) we have that y* E d£(Af)(y) o (Af)(y) + (A/)*(j,*) < (y,y*) +e & f(x) + f*(A*y*) < (Ax,y*) + e = (x,A*y*) + e 1 *> A*y* G def(x) &y*E A*' (def(x)). We used the relation (Af)* = f* o A*, proved in Theorem 2.3.1 (ix). • The subdifferential of a convex function 89 Corollary 2.4.7 Let /i,..., /„ : X -> R (n € N) be proper functions. Suppose that for every i € l,n there exists x~i £ Aora.fi suc/t i/iaf (/lD • • • •/„)(!! +... + !„) = /i(li) + • " • + /„(x„). (2.38) Then for every e € E+ ae(/iD...D/„)(ii+ ••• + !„) = [J{^ei/l(^l) n---n9e„/„(l„) | £l,...,e„ > 0, £X + ...+£„ =£}. In particular 0(/iD... n/n)(xi + • • • + xn) = a/i(ii) n • • • n 5/„(in). Conversely, if dfi(x~i) Pi • • • (~1 dfn(xn) ^ 0, i/ien relation (2.38) is vafo'd. Proof. We apply Corollary 2.4.5 to the functions fi,- • • ,fn and Corollary n 2.4.6 to / : X ->• E defined by /(xi,... ,£„) := /i(xi) + • • • + fn(xn) and to the operator A : X" —> X defined by A(xi,..., xn) := X\ + • • • + xn. If x* £ 5/i(«i) (~1 • • • n dfn{xn), then Vi £ l,n, Vij G-X" : (xi-Xi,x*) Corollary 2.4.8 Let /i,/2 e A(X), xt e dom/i for i e {1,2} and x — xi +X2- Assume that (fiDf2)(x) = /i(x"i) + f2(x2) and fiDf2 is subdifferentiable at x. If f\ is Gateaux differentiable at x\ then f\Of2 is Gateaux differentiable atx and V(fiDf2){x) = V/^xj). Moreover, if X is a normed vector space and /i is Frechet differentiable at xi then /1D/2 is Frechet differentiable at x. Proof. By the preceding corollary we have that d(f\Of2) (x) = df\ (x~i) n 0/2(z2)- Let x* £ d(f1nf2)(x). Then (u,x*) < (haf2)(x + u) - (AD/aXaf) < fl(xi +u) + /2(x2) - fi{xi) - /2(x2) Using (2.39) we obtain that (fiOf2)'(x,u) = (w, V/i(a;1)) for every u € X, which shows that /iD/2 is Gateaux differentiable at x and V(/iD/2)(x) = V/!(li). Assume now that X is a normed space and f\ is Frechet differentiable at x\. From the preceding situation we have that /1IH/2 is Gateaux differ entiable at x and V(/iD/2)(5;) — V/i(afi). Using again Eq. (2.39), we have that 0 < (/id/2)(af + u) - (fiDf2)(x) - (u,V/i(ii)> < fi{xi + u) - /i(ii) - (u, V/i(ii)> Vu G X; hence /1CH/2 is Frechet differentiable at 2; and V(/iD/2)(af) = V/i(aFi). D In Section 2.6 we shall extend the results of Corollaries 2.4.6 and 2.4.7 to the case where the infimum is not attained in y and x, respectively. The following result establishes a sufficient condition for the subdifferen- tiability of a convex function; this result is of exceptional importance. Theorem 2.4.9 Let f G A(X). If f is continuous at a; G dom/, then def(x) is nonempty and w* -compact for each e G M+. Furthermore, for every e > 0, f'e(x, •) is continuous and V«6l : f'e{x,u) = max{(«,x*) | x* G d£f(x)}. (2.40) Proof. Suppose that / is continuous at x G dom/ (from Theorem 2.4.4 we already know that def(x) ^ 0 for e > 0!). Let rj > 0. Since / is continuous at x, there exists V G Kx such that Vxex + V : f{x) Therefore (aT + V) x [f(x) + n, oo[C epi/, whence int(epi/) ^ 0. The set epi/ being convex and (x, f(x)) £ int(epi/) (because (x,f(x) — 5) $. epi/ for every S > 0), we can apply a separation theorem; so, there exists (a;*,a) G X* x E\ {(0,0)} such that V(z,f)€epi/ : (x,x*) + at < (x,x*) + af(x). (2.42) Taking x — x and t = /(aT) + n, n G N, we get a < 0. If a = 0, using Eq. (2.42), we obtain that {x — x,x*) < 0 for every x G dom/, and so, by Eq. (2.41), we have that (a;,a;*) < 0 for every x GV; since V is a neighborhood of the origin, this implies that x* = 0. Thus we obtain the contradiction: The subdifferential of a convex function 91 (a;*,a) = (0,0). Therefore a < 0; so we can consider a = -1 in Eq. (2.42). This relation becomes Vxedom/ : (x,x*) - f(x) < (x,x*) — f(x), i.e. x* e df(x). Let now e > 0. For every x* G dsf(x) = df£(x, -)(0), using Eq. (2.41), we have Viigy : (u,x*) def(x) C (fa + e)-1 V)° = (r, + e)Va. (2.44) By the Alaoglu-Bourbaki theorem (Theorem 1.1.10) we have that V° is w*- compact; since def(x) is w*-closed, it follows that d£f(x) is w*-compact. Taking into account that x £ int(dom/), from Theorem 2.1.14, we have that domf^x, •) = X; hence, using Theorem 2.2.13 and Eq. (2.43), the function fg(x,-) is continuous. Furthermore, using again Eq. (2.43), we have that f'e(x,u) > sup{(u,a;*) | x* £ def(x)}. We intend to prove that in the above inequality we have equality and the supremum is attained. For this end let u € X \ {0}. Consider XQ := Eu and ip : Xo -> E, p(tu) := tf'e(x,u); it is obvious that ip(u') < f'E(x,u') for every u' 6 Xo, and so, applying the Hahn-Banach theorem, there exists x* 6 X' such that (u,x*) = tp(u) = f^{x,u) and (u',x*) < f'e(x;u') for every u' € X. Since f^(x,-) is continuous, x* is continuous, too; hence x" e df'£{x, -)(0) = dEf(x). The proof is complete. • Using the preceding result we obtain the following criterion for the Gateaux differentiability of a continuous convex function. Corollary 2.4.10 Let f € A(X) be continuous atxe domf. Then f is Gateaux differentiate at x~ if and only if df(x) is a singleton. Proof. The necessity was already observed in Theorem 2.4.4(i). Assume that df(x) = {x*}. Using Theorem 2.4.9 we obtain that f'(x,u) = (u,x*) for every u € X. Because lim^o*-1 (f(x + tu) — f{x)) = —f'(x,—u) = 92 Convex analysis in locally convex spaces — (—u,x*) = (u,x*), we have that / is Gateaux differentiable and Vf(x) = x*. D Related to Frechet differentiability of convex functions see Theorem 3.3.2 in the next chapter. When / is not continuous at x £ dom/, relation (2.40) may be false; see Corollary 2.4.15 and Exercise 2.30. However the following result holds. Theorem 2.4.11 Let f £ T(X), x £ dom /, and e £ ]0, oo[. Then Vu£X : f'e(x,u)=sup{{u,x*)\x* £d£f(x)}. (2.45) Therefore f'e(x, •) is a Isc sublinear function. Furthermore, for every u £ X, f'(x,u) = lim (swp{{u,x*) | x* £ def(x)}) = inf (sup{(u,a:*) | x* £ def(x)}). (2.46) Proof. In Theorem 2.4.4(i) we have seen that dEf(x) — df^.(x,-)(0), whence V/6 3E/(f),Vii6l : {u,x*) < ft(x,u). Therefore the inequality " > " holds in Eq. (2.45). For proving the converse inequality, let u £ X and A £ R with A < f'£(x,u). From the definition of f'e (x, u), we have that Vt>0:X A < lim f^ + tu)-f(x)+e = ^ f(x + tu)-m = /oo(u), (2.48) t—>oo t (—>oo t Let us consider A := epi/ and B := {(x + su, f(x) + Xs — e) \ s > 0}. It is obvious that A and B are nonempty closed convex sets and, from Eq. (2.47), A n B = 0, i.e. (0,0) ^ A — B. Moreover B is locally compact (as subset of a finite dimensional separated locally convex space). Since Aoo = epi /oo and B00=R+- (u, A), from Eq. (2.48) we obtain that 4oonBco = {(0,0)}. Then, by Corollary 1.1.8, A — B is a closed set. Applying Theorem 1.1.7, there exists (x*,a) 6 X* x E such that V(x,t) £ epi/, Vs> 0 : 0 > (x - x- su,x*) + a(t - f(x) - sX + e). The sub differential of a convex function 93 Taking x = x and letting t -> oo we obtain that a < 0; if a = 0, for s = 0 we get the contradiction 0 > 0. Therefore a < 0 and we can suppose that a = — 1. The above relation becomes VxGdom/, Vs>0 : 0 > (a; -x,x*) - (/(«) - /(a?) +e) +s(A - (u,i*». For s = 0 we obtain that x* G dsf(x), while for s -> oo we obtain (u,a7*) > A. Therefore A < sup{(u,a;*) | a;* £ 5e/(aT)}. Since A < f^(x,u) is arbitrary, the relation (2.45) is true. The equality (2.46) follows from relation (2.45) and Theorem 2.1.14. • The subdifferentiability criterion of Theorem 2.4.9 can be extended. Theorem 2.4.12 Let f G A(X) and X0 := aff(dom/). // f\Xo is con tinuous atxG domf, then df(x) ^ 0. In particular, if dim X < oo, then df(x) ^ 0 for every x G *(dom/). Proof. Without any loss of generality, we can suppose that x = 0; then X0 = lin(dom/). The function g := f\x0 is convex, proper and continuous at 0. By Theorem 2.4.9 we have that dg{0) # 0. Let v? G dg{0); hence ip € XQ and Vxgdomj : tp[x) — (p(0) < g(x) — g(0). Using the Hahn-Banach theorem we get i* G X* such that x*\x0 = tp. The above inequality shows that Vx G dom/ = dome? : {x - 0,x*) = tp{x) < g(x) - g(0) = f(x) - /(0), whence x* G 9/(0). If dimX < oo, then dimXo < oo. By Theorem 2.2.21, g is continuous on int(dom/) = l(dom/). The conclusion follows from the first part. • Note that under the conditions of Theorem 2.4.12 df(x) is, generally, not bounded, and so it is not «;*-compact. But, under the conditions of Theorem 2.4.9, in the case of normed spaces, df has supplementary properties (mentioned in the next theorem). The local boundedness of monotone operators was proved by R.T. Rockafellar (see Theorem 3.11.14); the converse is true in Banach spaces for lower semicontinuous functions as shown by Corollary 3.11.16. Theorem 2.4.13 Let X be a normed space, f G A(X) be continuous on int(dom/) and e > 0. Then dsf is locally bounded on int(domc?/) = 94 Convex analysis in locally convex spaces int(dom/). Moreover, if f is bounded on bounded sets then f is Lipschitz on bounded sets and dsf is bounded on bounded sets. Proof. By Theorem 2.4.9 we have that int(dom/) C dome?/ C dom/; hence int(dom<9/) = int(dom/). Let xo £ int(dom/); since / is continuous at x0, from Corollary 2.2.12 we get the existence of m, p > 0 such that Vx,y€D(x0,2p) : \f(x) - f(y)\ < m\\x - y\\. Let us fix a; € D(xo,p). Then for y £ x + pUx we have that f(y) < f(x) + mp, i.e. Eq. (2.41) holds with V := pUx and r] :— mp. Using Eq. (2.44), we obtain that def(x) C (r? + e)V° = (m + e/p)Ux* for every x £ D(x0,p). Assume now that / is bounded on bounded sets. Then dom/ = X. Taking p > 0, / is bounded above on 2pUx, and so Eq. (2.41) holds with V := 2pUx and some r\ > 0. By Corollary 2.2.12 / is Lipschitz on pUx, 1 and, by Eq. (2.44), def(x) C (?? + e)p~ Ux* for x e D{x0,p). The proof is complete. • For other relationships between the continuity of / and the local bound- edness of df see Corollary 3.11.16. In Theorem 2.4.4 we have seen that finding the e-subdifferential of a convex function at a point reduces to compute the subdifferential of a sub- linear function at the origin. Other subdifferentials (the subdifferentials of Clarke, of Michel-Penot, etc.), for nonconvex functions are introduced through certain sublinear functions. So, we consider it is worth giving some properties of sublinear functions. Theorem 2.4.14 Let f,g:X->M.,h:Y->Rbe sublinear functions, T £ &(X, Y) and B,C C X* be nonempty sets. Then: (i) /* = ^9/(0); (ii) 5/(0) ^ 0 o / is Isc at 0; (hi) for every x G dom / and for every e £ R+ we have: dm = {x*edf(p)\(x,x') = f(x)}, d£f{x) = {x* e 3/(0) I (x,x°) > f(x) - £}, def(0) = 0/(0); (iv) if f is Isc at 0 then Vxel : J(x)=sup{{x,x*)\x* edf{0)}. The subdifferential of a convex function 95 (v) Suppose that f and g are Isc. Then f < g <3> 9/(0) C dg(0). (vi) The support function SB of B is sublinear, Isc, SB = (<-B)* and 0SB(O) — coB, the closure being taken with respect to the weak* topology w* on X*; moreover, SB < sc if and only if B C coC. (vii) // h is Isc then d(h o A)(0) = w*-c\ (A*{dh{0))); (viii) if f and g are Isc, then d(f + g)(Q) = w*-c\ (3/(0) + dg(0)). Proof, (i) For every x* € X* we have f*(x*) = sup^O - f{x) I x € dom/} > (0,a;*) - /(0) = 0. If a;* £ df(0) then (x,x*) < f(x) for every x £ dom/, whence f*(x*) = 0. If x* £ <9/(0), there exists x e X such that (x, x*) > f(x). In this situation we have f*{x*) > sup{(tx,x*) - /(tx) | t > 0} = sup{i(x,a;*) - /(x) | t > 0} = oo. (ii) If 9/(0) ^ 0 then, from Theorem 2.4.1(h), we have that / is Isc at 0. Suppose that / is Isc at 0. Then /(0) = /(0) = 0, and so, by Theorem 2.3.4, /(0) = /**(0) = 0. Assuming that <9/(0) = 0 we obtain that /* = £9/(0) = +00, and so the contradiction /** = — 00. Therefore 9/(0) ? 0. (iii) Let x £ dom/ and e > 0. If x* £ 9/(0) and (a;,a;*) > /(a;) — e then WxeX : {x-x,x*) = {x,x*)-(x,x*} *(l/,**) < /(i + ty) - f{x) +e< fix) + tfiy) - fix) + e = tf(y) + s. Dividing by t > 0 and letting then t -> 00, we obtain that (y,x*) > fiy) for every y € X, i.e. x* £ 9/(0). For e = 0 we have 0/(3:) = {x* € 0/(0) I (a;,a:*) > fix)} = {x* £ 0/(0) | (x,x*) = fix)}, while for x = 0 we obtain that 9£/(0) = K G 9/(0) I (0,ar*) > /(0) -e} - 9/(0). 96 Convex analysis in locally convex spaces (iv) Suppose that / is lsc at 0. Then /(0) = /(0) = 0. Using Theorem 2.3.4, we have that /** = J. Therefore J(x) = sup{(a;,a;*) - f*(x*) | x* 6 X*} = sup{(x,x*) \ x* € 0/(0)}. (v) It is obvious that 0/(0) C dg(0) if / < g. The converse implication follows immediately from (iv). (vi) At the end of Section 2.3 we noted that SB is lsc, sublinear and sB = {LB)*'• From the obvious inclusion B C 8SB(0) we get coB c <9SB(0) (because <9SB(0) is convex and w*-closed). Let x* fi coB. Using Theorem 1.1.5 in the space (X*,w*) and taking into account that (X*,w*)* — X, there exist x E X and AsK such that Vx* G coB D B : (x,x*) > A > (x,x*), whence (x,x*) > ssi'x), i-e. x* £ 8SB(0). The conclusion follows. (vii) By Theorem 2.4.2 we have that A*(dh(0) C d(h o A)(0). Let x* i w*-c\ (A*(dh(0))). Using Theorem 1.1.5, there exist x € Y and A e R such that (x,x*) > A > (x,Ay*) — {Ax,y*) for every y* e dh(Q). From (iv) we get A > h(Ax), and so x* $. d(h o A)(0). (viii) Let H : X x X -> 1, H(x,y) := f(x)+g(y), andT:I->XxX, Tx := (x,x). It is obvious that H is sublinear and lsc, and T is continuous and linear; moreover, dH(0,0) = 5/(0) x dg{0) and T*(x*,y*) = x* + y*. By (vii) we obtain that d(f + g)(0) = d(H o T)(0) = «>*-cl (T*(dH(0,0))) = w*- cl(5/(0) + dg(0)). The proof is complete. • Using the preceding result we have Corollary 2.4.15 Let f G \(X) and x G dom/. Then df(x) ^ 0 if and only if f'(x, •) is lsc at 0. In this case f'{x,-)(u) = sup{(u,x*) | Z* G df(x)} Vuel. Proof. The conclusion follows from the formula df(x) — df'(x, -)(0) (see Theorem 2.4.4) and assertions (i) and (iv) of the preceding theorem. • Note that, even for X = E2 and / G T(X) subdifferentiable at x, f'(x,-) may not be lsc; take for example / := i\j and x = (0,1), where U:={(x,y)em2 \x2+y2 Corollary 2.4.16 Let (X, || • ||) be a normed space, f : X -> R, f(x) := \\x\\, andx G X. Then, denoting Ux* by U*, we have: r = iu; df(0) = ir, df(x) = {x*eU*\(x,x*) = \\x\\}. Proof. The above formulas follow immediately from Theorem 2.4.14. • Another example of sublinear function is given in the following result. n 1 Corollary 2.4.17 Let / : E ->• E, f(x) := xi V • • • V xn, and x e W , where n G N. Then 5/(0) = An, /* = IA„, def(x) = {y e An\xiyi + ...+xnyn> f(x)-e}, where B A»:={(Ai,...,An)6" I Ai>0, Ai + ... + A„ = 1}. Proof. It is obvious that / is sublinear and continuous. Moreover y G <9/(0) »VxGC : xlVl + ••• + xnyn < n V • • • V xn. Taking Xj = 0 for j ^ i and Xi = —1, we obtain that — j/» < 0, i.e. yi>0 for every i. Taking then X{ — t G K for every i, we obtain that £(j/H \-yn) < t for every i, whence y±-\ \-yn = l. Therefore 9/(0) C An. The converse inclusion is immediate. The other relations follow directly from Theorem 2.4.14. • The following theorem furnishes a formula for the subdifferential of a supremum of convex functions. Other formulas will be given in Section 2.8. Theorem 2.4.18 (loffe-Tikhomirov) Let (A,T) be a separated compact topological space and fa : X —> E be a convex function for every a G A. Consider the function f := supaGAfa andF(x) := {a G A | fa{x) = f(x)}. Assume that the mapping A3a4/„(i) G E is upper semicontinuous and XQ G dom / is such that fa is continuous at XQ for every a £ A. Then df(x0)=co(\J dfa(x0)). (2.49) Proof. Since A is compact and a H-> fa{x) is upper semicontinuous we have that F{x) is a nonempty compact subset of A for every x G X. The inclusion D being true without any condition on A and / (and easy to prove), let us show the converse inclusion. If f(xo) — —oo, then fa(xo) = 98 Convex analysis in locally convex spaces —oo for all a £ A = F(x0), and so df(x0) = dfa(x0) — 0; the conclusion holds. We assume now that f(x0) £ R. Let us note first that in our conditions x0 £ (dom /)'. Indeed, let x £ X. Consider a fixed 70 > f(xo) (> fa(xo))- Because fa is continuous at xo, for every a £ A there exists ta > 0 such that fa(xo + tax) < 70. Since /? t-> /^(^o + tax) is upper semicontinuous, the set Aa := {/3 £ A \ fp(x0 + £Q:r) < 70} is an open set containing a. As A = UaeA-^<*> there exist ai,..., an £ A such that A = \J™=1 Aai. Let t := min{£ai,..., iQn } > 0. It follows that fa(x0 + tx) < 70 for every a £ A, and so f(xo + tx) < 70. Hence x0 £ (dom/)\ Since fa is continuous at xo, dfa(xo) 7^ 0 for every a £ F(xo), and so Q ^ 0, where Q is the set on the right-hand side of Eq. (2.49). Suppose that there exists x* £ df(x0) \ Q- Using Theorem 1.1.5 in the space (X*,w*), there exist x £ X and e > 0 such that VaeF(io),Vi'e9/a(i0) : (x,x*) - e > (x,x*), (2.50) or equivalently, by Theorem 2.4.9, VaeF(i„) : (x,x*)-e>{fa)'{x0;x). (2.51) 1 Because x0 £ (dom/) , we may suppose that x0+x £ dom/. Let t £]0,1[; then xo + tx £ dom /. There exists at £ A such that fat (XQ + tx) — f(x0 + tx). Because (1 - t)fa,(x0) + tfat(x0 + x) > fat(x0 + tx) and x* £ df(x0), we obtain that (1 - t)fat (Xo) > f(x0 + tx) - tfat (x0 + X) > f(x0 + tx) - tf(x0 + x) > f(xo) - t (x,x*) - tf(x0 + x). x It follows that limt^o fat( o) = f(xo). The space (A,T) being compact, there exists a convergent subnet (a^^j^j of (at)te]0,i[ converging to ao £ A; hence fao(x0) = f(x0), i.e. a0 £ F(x0)- Let s e]0,1[ be fixed (for the moment) and take t £ ]0, s]. Using Eq. (2.50) we obtain that fat(xo +SX) - fat(x0) fqt(x0 + tx) - fgt{x0) f(x0 + tx) - f (x0) s - t - t > (x,x*). We have that tp(j) < s for j y js for some js £ J. Taking t = ip(j) with The general problem of convex programming 99 j y js and using the upper semi-continuity hypothesis, we obtain that Va€]0>i[: Ui^ + ^-Ui^) s and so (jao)'{xo;x) > (x,x*), contradicting Eq. (2.51). D Of course, for conjugates and for the e-subdifferentials it is desirable to dispose of numerous formulas. There exist effectively a large set of such formulas we shall establish in Section 2.8. 2.5 The General Problem of Convex Programming By problem of convex programming we mean the problem of mini mizing a convex function / : X —> R, called objective function (or cost function) on a convex set C C X called the set of admissible solutions, or set of constraints. We shall denote such a problem by (P) min f(x), xeC. Of course, to consider this problem, X must be a linear space, however most of the results will be obtained in the framework of separated locally convex spaces or even normed spaces. To problem (P) we can associate a problem (apparently) without con straints: (P) min f(x), x G X, where / := / + ve in order for the problem (P) to be nontrivial, it is natural to assume that C n dom / ^ 0 (<£>• dom / ^ 0) and that / does not take the value — oo on C (i.e. f does not take the value — oo). We call value of problem (P) the extended real v(P) := v(f,C) := m£{f(x) | x G C} £ I; we call (optimal) solution of problem (P) an element x G C with the property that f(x) = v(P); this means that x is a (global) minimum point for the function /. We denote by S(P) or S(f, C) the set of optimal solu tions of problem (P). Therefore S(P) = {x G C | Vx G C : f(x) < f(x)} = {zGX|VxGX : f(x) < f(x)} = S(P) 100 Convex analysis in locally convex spaces if C fl dom / ^ 0. The set S(f, X) is denoted also by argmin /. Of course, an important problem is that of the existence of solutions for (P), resp. (P). The most important result which assures the existence of solutions for (P) is the famous Weierstrass' theorem. Because the underly ing spaces are not compact we have to use some coercivity conditions. We say that / : X —> E is coercive if limn^n^oo f(x) — oo. It is obvious that / is coercive if and only if all the level sets [/ < A] are bounded (see also Exercise 1.15); when / is convex then / is coercive if and only if the level set [/ < A] is bounded for some A > inf / (see Exercise 2.41). We have the following result. Theorem 2.5.1 Let f £ T(X). (i) If there exists A > v(f,X) such that [f < A] is w-compact, then s{f,x)^d>. (ii) If X is a reflexive Banach space and f is coercive then S(f,X) ^ ill. Proof, (i) Of course, v(f,X) = v(f, [f < A]). Since / is lsc and convex, / is tu-lsc. The conclusion follows using the Weierstrass theorem applied to the function /|[/ [f < /On)] and [f < f(x0)] + rUx C [/ < /fa)], where Ux := {x € X | IMI < 1}- Then X is reflexive. Proof. Of course, f(x0) < f(xi); if f(x0) = f{xi) then the relation [/ < f(xo)] + rUx C [/ < f(xi)] contradicts the boundedness of [/ < The general problem of convex programming 101 f(xi)]. Hence f(x0) < f(xi). Since / is Lipschitz on [/ < f(x\)], there exists L > 0 such that \f(x) - f(x')\ < L\\x-x'\\ for all x,x' £ [f < f(x\)]. Since /|[/symmetric set with nonempty interior, and so 0 € int A. Therefore there exist a, /? > 0 such -1 that allx C A c /SC/x- It follows that a 1|-|| = paux > PA > Ppux = P~x ||-||, where p^ is the Minkowski gauge associated to A. By Theorem 1.1.1 PA is a semi-norm, and so it is a norm equivalent to ||-||; moreover, by Proposition 1.1.1, the unit closed ball with respect to PA is cl A To obtain that X is reflexive, by the famous James' theorem (see [Diestel (1975), Th. 1.6]), it is sufficient to show that every x* € X* attains its infimum on A. Let x* £ X* \ {0} and a* := inf{(x,x*) \ x £ S}. Because S is bounded, a* € E. Let H := {x £ X \ (x,x*) = a*}. It is obvious that (S+eUx)nH # 0 for every e > 0. Taking e 6 ]0, S0] and xe £ (S+eUx)r\H, we obtain that f(xe) < f{x2) + eL, and so mixen f(x) < f(x2). By (ii) there exists xi £ H such that f(x~i) = infx€ij f(x), and so x\ £ Sf)H. Therefore (xi,x*) < (x,x*) for every x £ S. Similarly, there exists x2 £ S such that (x2,x*) > (x,x*) for every x £ S, whence there exists x := Xi —x2 £ A such that (x,x*) < (x,x*) for every x £ A. • Also the coercivity condition in Theorem 2.5.1(h) is essential as the next theorem will show. In order to establish it we introduce some preliminary notations and results. Let / 6 T(X) and denote by 11/ the set {g £ T(X) | domg = dom/, supx€domf \f(x) - g(x)\ < oo, inf / = inf g}. Since for g £ Hf we have that / — 7 d : Ilf x Ilf ^ M+, d{gug2) := supx6dom/ \f{x) - g{x)\ . Lemma 2.5.3 The mapping d is a metric on 11/ and (11/, d) is a complete metric space. Proof. It is obvious that d is a metric. Let (gn)n>i C 11/ be a Cauchy sequence. Then (gn{x))n>1 C E is a Cauchy sequence, and so (gn(x)) -> 102 Convex analysis in locally convex spaces g(x) E R for every x G dom/. Moreover, (supxedom/ \gn(x) - g(x)\)n^,1 -> 0. We extend g to X setting g(x) := oo for x G X \ dom /; hence dom g — dom /. Because (gn(x)) —> g(x) for every a; G X, from Theorem 2.1.3(h) we have that g is convex. Let e > 0; then there exists nE such that <7„(a;) — s < x g( ) < 9n(x) + £ for all n > nE and a; G dom/. It follows that g is proper (being minorized by gnc — e even on X) and inf / — e = inf gn — e < inf 0 which shows that inf g = inf / (even if inf/ = -co). We must only show that g is lsc. Take first x G domg = dom/ and A < g{x). There exists e > 0 such that A < g(x) — 2e. As above, there exists n = ne such that g(y) — e < gn(y) < g(y) + £ for every y G dom/. In particular g(x) — e < gn(x). Because gn is lsc at x, there exists U G Mx{x) such that Because gn is lsc at a; and A +1 < gn(x) = oo, there exists C/ G Mx{x) such that A + 1 < gn(y) for every j/ G U. It follows that A < gn(y) - 1 < 0 andy E X. Then the se£co(epi/U{(y,inf / — e)}) is i/te epigraph of a function fyi£ E T(X) with inf fViS = inf / — e and argmin /y>£ = y + K. Moreover, if f{y) < inf / + e then f — 2e < fy<£ < f, and so g := /yi£ + e G 11/ and d(f,g) < e. Proof. Consider a := inf / — e < inf /. Denote A := co (epi / U {(y, a)}) and let (x, t) E A and s > t. We want to show that t > a and (a;, s) E A. If these happen then epiipA = ^4 [VA being defined in Theorem 2.1.3(iv)], r fy,e := VA £ (X), /y,£(y) = a = inf /j,)£ and fy>e < /. Moreover, if f(y) < inf / + e, then f(y) — 2e < a, which shows that (y, a) E epi(/ — 2e). As / — 2e < f, we obtain that A C epi(/ - 2e), and so / - 2e < /y)£. Indeed, there exist the nets (Ai)igj C [0,1] and ((xi,ti))i€l C epi/ such that (a;,*) = lim,e/ (\i(xi,U) + (1 - Aj)(2/,a)), and so x - limi6/ (XiXi + (1 — Aj)y) and t = limjg/ (A;£; + (1 — Aj)a). Since [0,1] is a compact set, we may suppose that (A;)ie/ -> A G [0,1]. Of course, A;£; + (1 - Xi)a > Ajinf / + (1 - \i)a > a for every i E I, and so i > a. If (x,i) = {y,a) n then limn^oo (^(x0, to +ns- na) + ^{y, a)) = (x, s), and so (x, s) E A, The general problem of convex programming 103 where (xo,to) is a fixed element of epi/. In the contrary case Aj > 0 for 1 i >z_ j0 for some i0 £ I. Then limi^io (Xi(xi,ti+X^ (s-t))+ (1-Xi)(y,a)) = (x, s), and so (a;, s) £ A. Let u £ K; then (u,0) £ (epi/)oo- Fixing (xo,io) £ epi/, we have that (x0 + nu,io) £ epi/ for every n € N, and so ^(zo + nu, to) + ^^{y, a) £ A for every n £ N. Taking the limit we obtain that (y + u, a) £ A, whence fy Theorem 2.5.5 Let X be a reflexive Banach space and f £ T(X) be such that K n -K - {0}, where K := {u £ X | /<»(«) < 0}. Then the following statements are equivalent: (i) / is not coercive, (ii) there exists g £Uf such that argming = 0, (iii) {g &Uf \ argming = 0} is a dense G$ set (see page 34)- Proof. It is obvious that (iii) => (ii). (ii) =>• (i) Let g SUfbe such that argming = 0. From Theorem 2.5.1(h) we have that g is not coercive. Since g £ Uf, there exists 7 > 0 such that 9 — 7 < / < 5 + 7> which implies that / is not coercive, too. (i) =*- (iii) First of all note that for any g £ Uf, g is not coercive and /oo = Soo (because /-7 Gn •= {g e Uf I minxenC/x g(x) > inf g = 0}. The use of min instead of inf is possible because g is w-lsc and nllx is w-compact. Also note that Qn is open in (Uf,d); just take g € Qn and 104 Convex analysis in locally convex spaces 0 < 5 < mixenUx g(x); then the ball B(g,8) C Qn. As Q = f]n€^Gn, it is sufficient to show that Qn is dense for any n > 1. For this fix n > 1, e > 0 and /i G 11/. We may assume that /i(0) < e (otherwise replace /i by h(xo + •)> where xo is taken such that h(xo) < e). There exists y G X such that h(y) < e and (y + if) n nUx = 0- Indeed, when K = {0} take y € X \ nUx such that h(y) < e; this is possible because the set [h < e] is not bounded (see Exercise 2.41). Assume now that K ^ {0}. Then there exist z G K and r > 0 such that (z + if) n rUx = 0, r or equivalently z ^ r[/x — K. Otherwise K C flr>o ( Ux — K) C —K, a contradiction. The last inclusion is obtained as follows: consider z e r Hr>o ( Ux - K)\ then z = ^un — kn with un e Ux and kn € if, for every n € N, and so (fc„) ->• —z € if. Consider g := /iy]£ +e. By Lemma 2.5.4 we have that g G 11/, d(/i, g) Note that the reflexivity of the space was used in the proof of the pre ceding theorem only to ensure that the infimum of g on nUx is attained; so the preceding result remain valid when working on the dual of a normed space, the considered convex functions being w*-lsc. Also note that the condition Xo := if fl—if = {0} in the preceding theorem is not essential; if Xo 7^ {0} one obtains a similar result taking into account the constructions in Exercise 2.24. For a similar result when X is not a normed space see Exercise 2.25. Another important problem in optimization theory is the uniqueness of the solution when it exists. The following result gives an answer to this problem. Proposition 2.5.6 Let f G A(X). Then S(f,X) is a convex set. Fur thermore, if f is strictly convex then S(f, X) has at most one element. Proof. Let x G S(f,X); then S(f,X) = [f < f(x)], whence S(f,X) is convex. Let now / be strictly convex, and suppose that S(f,X) contains (at least) two distinct elements xi and xi\ since / is proper, S(f,X) C dom/. Then we obtain the contradiction v(f,X) < f (\Xl + |x2) < i/On) + \f[x2) = v(f,X). Therefore S(f, X) has at most one element. • The general problem of convex programming 105 As we already know, the practical method for determining extremum points of a function is to determine the points which verify the necessary conditions then to retain those which verify the sufficient conditions. In convex programming we have a very simple necessary and sufficient condi tion for optimal solutions. Theorem 2.5.7 If f £ A(X), then x E dom / is a minimum point for f if and only if 0 E df(x). Proof. Indeed, f(x) < f(x) for every x E X if and only if x £ dom / and 0 < f(x) — f(x) for every x £ X, which means that 0 € df(x). O Therefore, in convex programming, the minimum necessary condition is also a sufficient condition. In optimization theory local optimal solutions also play an important role; if g : X —>• E, we say that x £ X is a local minimum (resp. local maximum) point if there exists V € Nx{x) such that f(x) < f(x) (resp. f(x) > f(x)) for every x £ V. The convex programming problems present a particularity. Proposition 2.5.8 Let f : X —• E be a convex function. (i) Ifx(z dom f is a local minimum point for f, then x is also a global minimum point; (ii) if x € dom / is a local maximum point for f, then x is a global minimum point for f. Proof, (i) By hypothesis there exists V £ Nx(x) such that f(x) < f(x) for every x € V. Suppose that there exists x £ X such that f(x) < f(x); therefore f(x) £ E. Since V £ J^x(x), there exists A £]0,1[ such that y := (1 - X)x + Xx £ V. Therefore /(5?) < f(v) = /((I - A)3F + Ax) < (1 - A)/(3F) + Xf(x), whence the contradiction f(x) < f(x). Therefore x is a global minimum point for /. c (ii) By hypothesis there exists U £ J^ x such that f(x + x) < f(x) for every x £ U. If f(x) = — oo, it is clear that x is a global minimum point of /. Suppose that f(x) £ E (hence / is proper because x € int(dom/)). Then Vx€U : f{x) = f{\{x + x) + \{x-xj) <\f{x + x) + \f{x-x) Therefore f(x) = f(x) > f(x) for every x € x + U € Nj(x). Hence a; is a global minimum point of /. D This result explains why in a convex programming problem we look only for global minimum points. In practical problems, solved numerically on computers, frequently it is not possible to determine the exact optimal solutions (because one works with approximate values). A simple example in this sense is the problem of minimizing the function / : E —» R, f{t) := (t — 7r)2. Taking into account this fact, the notion of approximate solution is proved to be useful. More precisely, if e e R+, we say that x £ C is an e- (optimal) solution of problem (P) if f(x) < f(x) + e for every x 6 C; we denote by SS(P) or Ss(f,C) the set of e-solutions of problem (P). It is obvious that when C fl dom //0 we have that Se(f, C) = Se(f, X); moreover, if / is proper and S£{f,C) ^ 0, then v(f,C) G R and Se(f,C) = {x £ C | f(x) < v(f, C) + e}. Related to e-solutions we have the following result. Proposition 2.5.9 Let f £ A(X), x e dom/ and £ € P. Then Se(f,X) is convex; the set Se(f,X) is nonempty if f is bounded from below. Fur thermore, x € Sc(f,X) if and only if 0 6 d£f(x). • 2.6 Perturbed Problems We shall see (especially) in the following two sections that it is very useful to embed a minimization problem (P) min f(x), x 6 X, in a family of minimization problems. In this section X, Y are separated locally convex spaces if not stated explicitly otherwise and / : X —> R. Let us consider a function $:IxF4l having the property that f(x) — $>(x, 0) for every x € X; $ is called a perturbation function. For every y £ Y consider the problem (Py) min $(x,y), x 6 X. It is obvious that problem (P) coincides with problem (Po). To remain in the convex framework, we assume that $ is a convex function. Of course, when / : X —> R is convex we can find a function $ having the demanded property: $(x,y) := f(x) for all (x,y) E X xY. For Perturbed problems 107 obtaining useful results one chooses adequate perturbation functions, as we shall see in the sequel. Let $:Ix74lbea convex function and h : Y -» E, h{y) = v(Py), its associated marginal function (see page 43); h is also called the value or performance function associated to problems (Py). As noted in Theorem 2.1.3(v), h is convex, while from Eq. (2.8) we have that dom/i = Pry(dom$). The problem (P) min $(z,0), x £ X, is called the primal problem; we associate to it, in a natural way, the following dual problem {D) max (-$*(0,2/*)), y* € Y*. It is obvious that (£>) is equivalent to the convex programming problem (!?') min $*(0,2/*), y* eY*. The equivalence has to be understood in the sense that the problems (D) and (£>') have the same (e-)solutions; moreover v(D') = —v(D) (of course, for a maximization problem the notions of (e-)solution, local solution and value are defined dually to those for minimization problems). It is nice to observe that (£>') and (P) are of the same type. In the following results we establish some properties which connect the problems (P), (D) and the function h. Theorem 2.6.1 Let $ : X x Y -> E and h : Y ->• E be the marginal function associated to $. Then: (i) h* (y*) = $* (0, y*) for every y* eY*. (ii) Let (x,y) € X x Y be such that ${x,y) 6 R. Then (0,j/*) € d$(x,y) O h(y) = $(x,y) and y* £ dh{y). (hi) v(P) = h(0) and v(D) = /i**(0). Therefore v(P) > v(D); in this case we say that one has weak duality. (iv) Suppose that $ is proper, x € X and y* € Y*. Then (0,y*) € d$(x~, 0) «/ and only if x is a solution of problem (P), y* is a solution of (D) and v(P) = v{D) G E. Assume, moreover, that $ is convex. (v) /i(0) £ E and /i is Isc at 0 <£> u(P) = u(D) £ E; in this case one says that we have strong duality. 108 Convex analysis in locally convex spaces (vi) h(0) G E and dh(0) / 0 & v(P) = v(D) G E and (D) has optimal solutions. In this situation S(D) = dh(0). (vii) Suppose that $ is proper. Then [ h is proper] •£> [ h* is proper] •£> [ h is minorized by an affine continuous functional] •& 3y* 6 7', 3a GE, V(x,y)eXxY : $(x,y) > (y,y*) + a. (2.52) Proof, (i) We have h*(y*) = sup({y,y*) - h{y)) = sup ((y,y*) - inf $(x,y)) y€Y y€Y \ ^^ / = sup sup((y,y*) -$(x,y)) = sup ((x,0) + (y,y*) - ${x,y)) y€YxeX (x,y)€XxY = **(<), y*). (ii) Assume that $(x,y) G E. Let (0,j/*) G d$(x,y). Then by (i) and Theorem 2.4.2 (iii), h(y) < *(x,y) = (x,0) + (y,y*) - $*(0,i/*) = (y,y*) - h*(y*) < h(y), and so h(y) = $(x,y) and y* G dh(y). Conversely, if these two conditions hold then *{x,y) = h(y) = {y,y*) - h*{y*) = (x,0) + (y,y*) - **(0,y'), whence, again by Theorem 2.4.2 (iii), (0,2/*) G d$(x,y). (iii) It is obvious that v(P) — /i(0); moreover, v(D) = sup (-$(0,r/*)) = sup ((0,2/*) -h*(y*)) = h**{0). y*EY* y*&Y* Therefore v(P) >v(D). (iv) If (0,|7*) G<9$(x,0) then v(P) = h(Q) < $(x,0) = -$*(0,2/*) < v(D) = x**(0) < h(0), whence x is a solution of (P), y* is a solution of (£>) and v(P) = v(D) G E. Conversely, if these last assertions are true, we have -$*(0,j7*) = h**(0) = h(Q) = S(z,0) G E, whence (z,0)Gdom$ and §(x,0)+ **(0,y*) = (x,0) + (0,y*), Perturbed problems 109 which shows that (0,y*) G d$(x,0). Assume now that $ is convex. (v) If h is lsc at 0 and h(0) G R we have that h(0) G E, whence h** = ~h. Therefore v(D) = h**(0) = ft(0) = v{P) G E. Conversely, if this last relation is true, then h(0) = h(0), i.e. h is lsc at 0. The conclusion holds. (vi) Suppose that /i(0) G E and dh(0) ^ 0; then /i is lsc at 0, whence, by (iv), we have that v(P) = v(D) G E. Let y* G dh(0). Then h(0) + h*(y*) = 0, whence Vy* G Y* : v(D) = v(P) = h(Q) = -h*(y*) = -**(<), IT) > -*'{0,y*). Therefore 0 ^ dh(0) C S(D). Conversely, suppose that v(P) = v(D) G R and that (D) has solutions. Let y* G S(D). Then ft(0) = h**(0) = -h*(y*) G E, whence y* G 9ft(0). Thus we have that 0 # S(D) C 0ft(O). (vii) The mentioned equivalences are obvious since h is convex, and dom/i^0. • We say that the problem (P) is normal if v(P) = v(D) G E; (P) is stable if w(P) = v(D) G R and (D) has optimal solutions. Theorem 2.6.1 above gives characterizations of these notions. In the following result we establish formulas for deh(y) and deh*(y*) when h is proper. Theorem 2.6.2 Let $ G A(X x Y) satisfy condition Eq. (2.52) and e G E+ . Then: (i) /or every y G dom h e y G deh(y) = fl,>0 LUjcfo* * I (°'^) ^+^0^)} 7)>0 *(x,y) (ii) /or every y* G dom/i*; cU*(y*) = p| cl{yGY|3xGX : (Q,y<) € de+v*(x,y)}. 110 Convex analysis in locally convex spaces Proof, (i) It is obvious that , n n {»*i(o,!/ )6flt+,^»)} V>0 $(x,y) Let y* G deh(y), i.e. h(y) + h*{y*) < (y,y*) +e, and let rj > 0 and a; € X be such that $(x,y) < h(y) + rj. Then $(x, 2/) + $*(0, y*) < ft(y) + r, + h*{y*) < (y, y") + e + r,, i.e. (0,2/*) € d£+n$(x,y). Therefore dsh(y) C f| f| {y*\ (0,y*) G &+„*(*, y)}, V>0 (ii) Let y € d£h*(y*) and r\ > 0. Then />**( It follows that for every V G N(j/) there exists yv G V such that Ml/v)+ &*(!/*) < From the definition of h we get xy E X such that $(xv,yv) + h*(y*) = $(xv,yv) + $*(0,y*) < (yv,y*) +e + rj, i.e. (0,y*) G de+n j/Gcl{t/Gy|3a;GX : (0,y*) G 3£+7?$(z,y)}. Taking into account that 77 > 0 is arbitrary, we obtain that d£h*(y*)cf) d{y\3xGX : (0,y') G de+r,*(x,y)}. Let y be in the set from the right-hand side and 77 > 0. Then, using the continuity of y*, there exists Vo G N(y) such that (z,y*) < (y,y*) + r)/2 for every z G VQ. On the other hand, for every V G N(y) there exist Perturbed problems 111 (xv,Vv) € X x Y such that yv GVTlVo and (0,y*) G de+r,/2$(xv,yv); hence Kw) + h*{y*) < ${xv,yv) + $*(0,y*) < (yv,y*)+e + i]/2 < (y,v*)+e+T]. Therefore VVeNfe) : inf %) + A*(y*)< (J/,!/*>+£ + »/, yev i.e. h(y) + h*(y*) < (y,y*) +E + TJ. Since TJ > 0 is arbitrary, we obtain that h"(y) + h*(y*) = h(y) + h*(y*) < (y,y*) + e, i.e.ytd£h*(y*). • The preceding result is useful for deriving formulas for e-subdifferentials for other functions. Theorem 2.6.3 Let $ E T(X x Y) and cp : X -» E, ip(x) := F(x,0). Then for every e E E+ and every x E dom ip we have dMx)=C\ w*-c\{x*eX*\3y*eY* : (x*,y*) E d£+v$(x,0)} = f| w*-d{x*eX*\3y*eY* : (x,0) E de+v^(x*,y*)}. Proof. Let us consider the spaces X*, Y* endowed with their weak* topologies and the function k:X*^W, k(x*):= inf $*(x*,y*). y*€Y* Then k* : X ->• I, A;*(a;) = $**(x,0) = $(x,0) = ip(x). Using Theorem 2.6.2, we have dMx) = dEk*(x) = fl w*-c\{x* | 3j/* : (a;,0) E &+-,**(**>!/*)} = f| W*-C1{Z*|3T/* : (x'.y'Jeft+^O)}. 1 >J1>0'77 X) The proof is complete. D Let us apply the results of Theorems 2.6.2 and 2.6.3 in some particular situations. Stronger conclusions, but under more restrictive conditions, will be established in Section 2.8. 112 Convex analysis in locally convex spaces Corollary 2.6.4 Let A G H{X,Y) and f : X —> R be a convex function for which 3y* £Y*, 3a G R, Vz e X : f(x)> (x,A*y*)+a. If (Af)(y) G 1 (O y G A(dom/) = domA/), j/* G dom(/* o A*) and e > 0, then de{Af){y) = fl [J A-\de+rif{x)) n>0 Ax=y = n n A*-\d£+rif(x)), »7>0 Ax=y,f(x)<(Af)(y)+r, and ds(f*oA*)(y*) = f]d{yeY\3xeX : Ax = y, A*y* G de+nf(x)}. Proof. Let us consider K *,y),in •= ' | oo if Axjty. Then $*(x*,t/*) = /*(x* +AV) and (O.y*) G 0„$(a;,y) The result follows immediately from Theorem 2.6.2. D Corollary 2.6.5 Let A G L(X, Y) and f G T(Y). Then d£(foA)(x) = O^lv'-dA^de+rffiAx)) for every x G A_1(dom/) = dom(/ o A) and every e > 0. Proof. Let us consider $:Ixy^l, $(x,y) := f(Ax+y), and ip(x) :— $(x,0) = f(Ax). Then$*(x*,y*) = f(y*) ii A*y* =x*, $*(x*,y*) = oo otherwise. Moreover (x*,y*)edv*(x,0)&A*y*=x' and f(Ax) + f*(y*) < (Ax,y*) + rj & A*y* = x* and y* G dvf(Ax). The conclusion follows from Theorem 2.6.3. • The fundamental duality formula 113 Corollary 2.6.6 Let /i, /2 € A(X) for which 3x* eX*, 3a el, \/xeX, Vie {1,2} : fi{x) > {x,x*) + a. If (hnf2)(x) e E and e > 0, then: de(fiDf2){x) = f) 1J (0ei/i(a; - y)ndeJ2(y)) n>0 yeX,Ei>0,e+n=ei+£2 = n n u (^x/i^-^n^/ad/)), ri>0yeSrl(x) £i>0,e+r)=£i+£2 x n d{huf2){x) = r\v>0Uy€X (W - y) ^/a(i/)). where Sv(x) := {y e X \ Mx - y) + f2{y) < {fiDf2)(x) + r?}. Proof. Let us consider / : X x X -)• E, f{x\,x2) := fi(xi) + f2(x2) and A e £(X x X,X), A(xi,x2) := £i + £2- The conclusion follows from Corollary 2.6.4. • Corollary 2.6.7 Let /i,/2 € T(X). // a; e dom/i n dom/2 and e > 0 then: ds{h+mi)=n ™*-d ( u (^Aw+aeaMx)) 7)>0 y£i>0,£+77=ei+e2 5(/i + f2)(x) = fl tiT-d^/iC*) + d„f2(x)). Proof. Let us consider /:IxI->I, /(^ii x2) := /i(xi) + 72(^2) and A e £(X, X x X), Ax := (x,x). The conclusion follows from Corollary 2.6.5. • 2.7 The Fundamental Duality Formula The following theorem is very useful for obtaining important results in convex programming; this is the reason for calling formula (2.53) the fun damental duality formula of convex analysis. Theorem 2.7.1 Let $ e A(X x Y) be such that 0 6 Pry(dom$). Con sider Yo := lin (Pry (dom $)). Suppose that one of the following conditions is satisfied: 114 Convex analysis in locally convex spaces (i) there exists Ao £ E such that VQ := {y £ Y \ 3x £ X, $(x,y) < AO}S:NVO(O); (ii) there exist Ao G E and x0 £ X such that VUeNx : {y£Y\3x£x0 + U, $(x,y) < Ao} e Kyo(0); (iii) there exists XQ £ X such that (a;o;0) £ dom$ and $(xo,-) is con tinuous at 0; (iv) X and Y are metrizable, epi $ satisfies condition (Hwz) on page 14 i6 and 0£ (Pry(dom$)); (v) X is a Frechet space, Y is metrizable, $ is a li-convex function and i6 0e (Pry(dom$)); (vi) X is a Frechet space, $ is Isc and 0 £ j6(Pry(dom$)); (vii) X, Y are Frechet spaces, $ is Isc and 0 € lc(Pry(dom$)); (viii) dimlo < 00 and 0 £ *(Pry(dom$)); (ix) there exists Xo £ X such that $(XQ,-) is quasi-continuous and the sets {0}, Pry(dom$) are united. Then either h(0) = —00 or h(0) £ E and h\y0 is continuous at 0. In both cases we have inf $(x,0)= max ( - **(0,i/*)). (2.53) x€X ' y*€Y' v " ' Furthermore, x £ X is a minimum point for $(-,0) if and only if there exists y* £ Y* such that (0,y*) £ d$(x,0). Proof. Since 0 £ Pry(dom$), h(0) < 00. If h(0) = —00 we have h*(y*) — °° f°r every y* 6 Y*, whence —$*(0, j/*) = -00 = h(0) for every y* £ Y*. Therefore the conclusion is true in this case. Let us consider now the case h(0) £ K. Suppose that condition (i) is verified. Then h\y0 is bounded above by Ao on VQ. Since h is convex, h\y0 is continuous at 0. By Theorem 2.4.12 we have that dh(0) f 0. Then relation (2.53) follows Theorem 2.6.1 (vi). (ii) =£- (i) This implication is obvious (just take U — X). (iii) => (ii) Suppose that (iii) holds. Since $(xo,-) is continuous at 0, the set V0 := {y | $(xo,y) < $(x0,0) + 1} is a neighborhood of 0 (in particular YQ = Y). Taking Ao := $(a;o,0) + 1, it is obvious that V0 C {y \3x £ x0 + U : $(x,y) < A0} for every U £ Nx- Therefore (ii) holds. The fundamental duality formula 115 (iv) => (ii) Suppose that (iv) holds. Let us consider the relation 31 : X xR=$Y whose graph is given by grtt := {(x,t,y) \ (x,y,t) € epi$}. From the hypothesis 51 satisfies condition (Hwi), whence, by Proposition 1.2.6(i), % satisfies condition (Hw(z,i)), too. Moreover 0 € i6(Im3?). Let (xo,io) e X x R be such that 0 6 Jl(xo,to)- Applying Theorem 1.3.5 we obtain that "R{{x0 + U)x ] - oo,t0 + 1]) £ ^y0(0) for every U eNX- This shows that (ii) holds with Ao := t0 + 1. (v) => (ii) By Proposition 2.2.18, there exist a Frechet space Z and a cs-closed function F : Z xX xY -^R such that $(x, y) = infzez F(z, x, y) for all (x, y) € X xY. The conclusion follows like in the preceding case by replacing X by Z x X, x by (z,x) and U by Z x U. (vi) => (ii) The proof is the same as for (iv) =$• (ii) with the excep tion that one uses Ursescu's theorem (Theorem 1.3.7) instead of Simons' theorem. It is obvious that (vii) implies conditions (iv), (v) and (vi). If (viii) is verified, the conclusion follows from Theorem 2.4.12. (ix) => (i) It is obvious that $(a;0, •) > h. By Proposition 2.2.15 we have that h is quasi-continuous. It follows that rint(dom/i) ^ 0. Using Proposi tion 1.2.8 we obtain that 0 £ rint(dom/i). Therefore h\y0 is continuous at 0, and so (i) holds. Of course, if x is a minimum point for $(-,0) then x is a solution of (P) (from p. 107); it follows that $(x,0) = v(P) = v{D) G R. Let y* be a solution of (D) (in our conditions a solution exists). Then, from Theorem 2.6.1(iv), (0,2/*) £ d$(x,0). The converse implication follows from the same result. • When the properness condition in the preceding theorem is violated relation (2.53) is automatically verified. Indeed, in this case there exists (xi,yi) £ X x Y such that $(xi,j/i) = -oo, and so h(yi) = —oo. Because 0 € '(dornh), by Proposition 2.2.5 we have that h(0) — -oo. In applications it is important to have conditions on $ which also ensure that the functions $, $(a;, y) := <&(x,y) — {x,x*) with a;* 6 X*, satisfy them. Such conditions are conditions (ii)-(ix) of Theorem 2.7.1, but this is not always true for (i). 116 Convex analysis in locally convex spaces Other conditions of this type are: 3A0 G R, 3B G %x : {y G Y | 3x G B, $(x,y) < A0} G >JVo(0), (2.54) V?7 G Kx, 3A > 0 : {yeY\3x£XU, $(x,y) < A} G Nyo(0), (2.55) where "Bx is the class of bounded subsets of X and, as in Theorem 2.7.1, Y0 =lin(Pry(dom$)). Proposition 2.7.2 Let $ G A(X x F). Conditions (i) — (viii) foez'ng those from Theorem 2.7.1, we have: (iii) =>• (2.54) =S> (2.55) «• (ii) => (i), (2.55) => (2.54) «/ X is a normed vector space, (vii) =>• (iv) A (v) A (vi), (iv) V (v) V (vi) =* (ii) and (viii) =* (2.54). Moreover, taking D = Pry(dom$), one has: if dim(linZ)) < oo t/ien *D = rint£>; i/ X, Y are metrizable, epi$ satisfies H(x) and lbD ^ 0 then thD = rintZ?; similarly for the situations corresponding to conditions (v)-(vii). Proof. The implications (vii) =>• (iv) A (v) A (vi), (iv) V (v) V (vi) =>• (ii) =*• (i) were already observed (or proved) during the proof of the preceding theorem. The implication (iii) =$• (2.54) is obvious; just take B — {XQ}. (2.54) =>• (2.55) Let U G Kx; there exists /x > 0 such that B C fill. Taking A = max{Ao,/i} we have that {yeY\3x£B, $(x,y) < A0} C {y G Y \ 3x G ^?7, $(a;,j/) < A0} CfeeY |3xG XU, 9(x,y) < A}. The conclusion follows. (ii) =>• (2.55) Consider A0 G M. and zo G X given by (ii). Let U G N*. There exists fi > 0 such that a;0 € /if/. Let V = {y€Y\3x£x0 + U, $(x,y) < Ao} G Ny0. Taking A = max{Ao,/« + 1} and y € V, there exists x G x0 + U such that $(x,y) < A0. As x G x0 + U C /J,U + U = (fi + 1)U C At/, the conclusion follows. (2.55) =>• (ii) It is obvious that there exists x0 G X such that $(xo, 0) < oo. Consider Ao = max{$(a;o,0),0} + 1 and let U G Nx- There exists Uo G Nx such that UQ + Uo C U. There exists also Ai > 0 such that XQ G Aif/0. By hypothesis, there exists A > Ao + Ai such that Vb = {y G y | 3a; G AE/0, #(a;,i/) < A} G Nyo(0). Let V = X^VQ and take yeV. As Xy G Vo> there exists x' G Af/o such that $(x',Xy) < X. It follows that 1 1 $ ((1 - A^Xzo.O) + A"V,Av)) < (l-A- )$(a;o,0)+A- $(a;',A2/) < A0, The fundamental duality formula 117 whence $ (x,y) < Ao, where x := xo + X~1(xl — Xo) G Xo + Uo + Uo C Xo + U. (2.55) => (2.54) when X is a normed vector space. Take U = Ux = {x G X | ||x|| < 1}. There exists A0 > 0 such that {y G Y \ 3x G X0U, ${x,y) < Ao} € Nyo(0). As XoU is bounded, the conclusion follows. i (viii) =$• (2.54) Suppose that dim Yb < oo and 0 £ (Pry(dom$)). It follows that there exist j/i,... ,ym G Pry(dom$) such that Vo = co{y\,..., 2/m} £ !Nyo(0). For every i € l,m there exists Zj G X such that (xi,yi) G dom$. Let Ao = max{$(a;j,y;) | 1 < i < m} and B — co{xi,..., xm}. It is obvious that B is bounded and for y G Vo there exist Aj,..., Am > 0 with Y!T=i ^i = 1 such that j/ = YlT=i ^'Vi- Then a; = YllLi ^ixi € -^ anc^ *(z,2/) < A0. The fact that lD — v'mtD if dim(lin£>) < oo is obvious. Let X, Y be lb metrizable, epi$ satisfy (Hx), and consider y0 G D. Taking ($(•,0))* (&") = min $*(£*,I denned by $(x,y) := $(x,y) — (x,x*). As observed above, the function $ satisfies the same condition as $ among those mentioned in the statement of the corollary. Applying Theorem 2.7.1 (and eventually the preceding proposition), we have that inix&x $(x,Q) = maxy.ey. (-$*(0, ?/*)). But $*(0,2/*) = $*(x*,y*), and so the conclusion follows. • We state another duality formula which will be useful in the sequel. Theorem 2.7.4 Let F G A(X x Y), 6 : X =t Y be a convex multi function, and D = U{G(x) — y \ (x,y) G domF}. Assume that 0 G D and let Y0 = linD. If one of the following conditions holds: (i) for every U G Nx there exist A > 0 and V G Ny0 such that {0} x V C grCn (XU xY)-[F< A]; 118 Convex analysis in locally convex spaces (ii) there exist Ao G K, B G T>x and VQ G Ny0 such that {0} x V0 C grC n (B x Y) - [F < A0]; (iii) there exists (xo,yo) G grC (~l domF such that F(xo, •) is continuous at y0; (iv) X, Y are metrizable, 0 6 lbD and either F is cs-complete and C is cs-closed, or F is cs-closed and C is cs-complete; (v) X, Y are Frechet spaces, F and C are li-convex, and 0 G lbD; (vi) dim YQ < oo and 0 € *£>, £/ien £/iere exists z* G Y* such that inf{F(x,y) \ (x, y) G gr 6} = inf{F(x,y) + (z, z*) \ (x,y + z) e gr e}. Proof. Let Z := Y and *:(IxZ)x74l, $(x,z;y) := F(x,z) + tgre(x,y + z). It follows easily that $ is convex and Pry(dom$) = D. It is obvious that a; 2: the conclusion of the theorem is equivalent to inf(X]2)exxZ^( , iO) = max^.gy. — $*(0,0;2/*). So we have to show that if one of the conditions of the theorem is verified then a condition of Theorem 2.7.1 holds. If (i) holds it is immediate that $ verifies condition (i) of Theorem 2.7.1. The implication (ii) => (i) is obvious. We also have that (iii) => (ii); just take B := {x0}, A0 := F(x0,yo) + 1 and V0 = {y G Y \ F(x0,y0+y) < A0} (in this case Y0 — Y). Similar to the proof in Proposition 2.7.2 we have that (vi) =>• (ii). x (iv) => (i) Let Y^n>i ^n( n, zn, yn, tn) be a convex series with elements of epi $ such that J2n>i ^nXn and J2„>i ^nzn are Cauchy, J2n>i Kyn = y G Y and J2n>i ^tn = t G K Then (xn,zn,tn) G epiF and (xn,zn + yn) G grC for every n > 1. If F is cs-complete it follows that J2n>1 Xnxn and Sn>i ^nZn are convergent with sums x G X and z G Z, respectively; moreover, (x,z,t) G epi.F, whence (x,z + y) G grC since grC is cs-closed. The same conclusion holds in the other case. Therefore $ verifies condi tion (H(x,z)) in our hypotheses. Hence condition (iv) of Theorem 2.7.1 is verified. So, by Proposition 2.7.2, relation (2.55) holds. Therefore for U x Y G Nxxz there exists A > 0 such that {y G Y | 3 (x, z) G XU x Y, $(x, z; y) < A} = {y&Y\3xG\U, 3zGZ : (x,y + z) G e, F{x,z) < A} G Nyo(0), The fundamental duality formula 119 and so (i) holds. (v) => (i) Consider the set A:={(x,z,y) eX x Z xY \y + z£ G(x)} = {{x,y' -y,y)\{x,y')egre, yeY} = grC x {0} + {0} x {(-y,y) \y eY}. Using Propositions 1.2.4 (ii) and 1.2.5 (ii) we obtain that A is li-convex (as sum of two li-convex subsets of a Frechet space). Since $(x,z;y) — F(x, z) + LA{X, Z, y) and the functions F and LA are li-convex, $ is li-convex, too. Thus $ satisfies condition (v) of Theorem 2.7.1; the other conditions being obviously satisfied, as in (iv) =4- (i), we get that (i) holds, too. • Note that every condition of the preceding theorem is verified by F, F(x,y) — F(x,y) — (x,x*), where x* € X*, when the same condition is verified by F. Taking gr 6 = X x {0}, the conclusion of the preceding theorem is just the conclusion of Theorem 2.7.1. Conditions (i), (ii), (iii) and (vi) become conditions (2.55), (2.54), (iii) and (viii) of Theorem 2.7.1, respectively; to conditions (iv) and (v) correspond slightly stronger forms of conditions (iv) and (v) of Theorem 2.7.1, respectively. Remark 2.7.1 If F(x,y) = f{x) + g(y) with / € A(X), g € A(Y), for condition (i) of Theorem 2.7.4 it is sufficient (and necessary if f,g have proper conjugates) to have VUelSx, 3A>0, 3V €Xy0 : V C [g < A] - e(\U n [/ < A]), while for condition (ii) of Theorem 2.7.4 it is sufficient (and necessary if /, g have proper conjugates) to have 3 A0 S 1, Be Sjr, V0 £ NYo : V0 C [g < A0] - G{B n [/ < A0]). Of course, these two conditions are equivalent if X is a normed space. Corollary 2.7.5 Let f € A(X) and A 6 &(X,Y). Suppose that one of the following conditions is verified: (i) / is continuous on int(dom/), assumed to be nonempty, and A is relatively open, i.e. A(D) is open in ImA for every open subset D C X; (ii) X and Y are metrizable, either (a) / is cs-complete or (b) / is cs-closed and gr A satisfies condition (Hx), and *6A(dom/) ^ 0; 120 Convex analysis in locally convex spaces (iii) X is a Frechet space, Y is metrizable, f is a li-convex function and ib A(domf)jt Then, either Af is —oo on *(^4(dom/)) or Af is proper and {Af)\y0 is continuous on ' (A(dom f)). Moreover Vj/G^dom/)) : (Af)(y) =max{{y,y*) - r(A*y*)\y* eY*}. l Proof. Let us consider y0 6 (A(dom/)) and $ : X x Y -*• I, $(ar,y) := f(x) + tgTA(x, 2/o + y). We have that Pry (dom $) = yl(dom/) - y0- If condition (ii), (iii), (iv) or (v) is verified, then $ satisfies condition (iv), (v), (vi) or (viii) of Theorem 2.7.1, respectively. Suppose that condition (i) is satisfied. (Obviously, it is impossible that condition (iii) of Theorem 2.7.1 be verified in this case: F(xo, •) = i{Ax0}-) Since A is relatively open we have that *(A(dom/)) = intim/i (A(dom/)) = A(int(dom/)) (Exercise!), whence j/o = AXQ for some Xo € int(dom/). It follows that / is bounded above on a neighborhood Vo of Xo, and so h (the marginal function associated to $) is bounded above by the same constant on J4(VO) — yo, which is a neighborhood of 0. Therefore condition (i) of Theorem 2.7.1 is verified. So, under each of the five conditions, we have that (Af)(y ) = inf $(a:,0) = max (-$*(0,y*)) = max «y ,y*>-/*(>lV)), 0 xex y*eY* yer* 0 doing similar calculations to those from Theorem 2.3.1(ix). • Corollary 2.7.6 Let f1}..., fn G A(X) and take f :- /iD • • • •/„. Sup pose that one of the following conditions is verified: (i) /i is continuous at some point in dom /i, (ii) X is a Frechet space, the functions /i,...,/n are li-convex and i6(dom/)^0, (iii) dimX < oo. ! Then either f is —oo on (dom/) or /|aff(dom/) is finite and continuous on t(domf), in which case f is sub differentiable on this set. Furthermore, for every x € '(dom/) we have that f(x) = max ((*, x*) - /•(*•) /•(*')) = (/; + ••• + rn)*(x). Formulas for conjugates and e-sub differentials 121 Proof. Recall that dom / = dom /i -\ + dom /„. In the cases (ii) and (iii) the result is an immediate consequence of Corollary 2.7.5 taking $ and A as in Corollary 2.4.7, while for (i) one does the proof by induction. • 2.8 Formulas for Conjugates and e-Subdifferentials, Duality Relations and Optimality Conditions In the preceding sections we have considered only situations in which it was simple to compute conjugates and e-subdifferentials: sum of functions with separated variables, convolution of convex functions, and functions of type Af. For the other types of functions, generally, it is more difficult to compute the conjugate functions or the e-subdifferentials. In this sec tion, we intend to establish sufficient conditions, as general as possible, in order to ensure the validity of such formulas. In this section the spaces X,Xi,..., Xn and Y are separated locally convex spaces if not stated ex plicitly otherwise. We begin with the following result. Theorem 2.8.1 Let F £ A(X x Y), A £ H{X,Y) and E, (i) there exist Ao £ K, Vo € Ny0 and B G "S>x such that {0} x Vo C {{x,Ax) \xeB}-[F< Ao]; (ii) for every U £ Nx there exist A > 0 and V £ Ny0 such that {0} x V C {(x, Ax)\xe At/} ~[F< A]; (iii) there exists xo £ X such that (:ro,j4a;o) £ domF and F(xo,-) is continuous at AXQ; (iv) X and Y are metrizable, 0 E tbD and either F is cs-closed and gr A is cs-complete or F is cs-complete; (v) X is a Frechet space, Y is metrizable, F is li-convex and 0 £ lhD; (vi) X is a Frechet space, F is Isc and 0 £ lbD; (vii) X and Y are Frechet spaces, F is Isc and 0 £ %CD; (viii) dimy0 < oo and 0 £ 'D, 122 Convex analysis in locally convex spaces (ix) there exists XQ £ X such that F(xo, •) is quasi-continuous and {0} and D are united- Then for x* £ X*, x £ domip and e > 0 we have: tp*(x*) = mm{F*(x* - A*y\y*) \ y* £ Y*}, (2.56) dM*) = {A*y*+x* | (x*,y*) € dEF(x, Ax)}. (2.57) Proof. It is easy to verify that Vx* £ X* : - Applying Theorem 2.7.1, we obtain that -V(a;*)=inf{S(x,0) | X e X} = max { -*'(0, -y") \y* £Y*}. (2.58) But **(0, -V*) = sup{(ar, a;*) + (y, -y*) - F(x, Ax - y) \ (x,y) £ X x Y} = sup{(a;,a;*) + (z - Ax,y*) - F(x,z) \ (x,z) £ X x Y} = sup{(a;, x* - A*y*) + (z, y*) - F(x, z) \(x,z)£Xx Y} = F*(x*-A*y*,y*). Prom Eq. (2.58) we get immediately Eq. (2.56). Note that the inclusion "D" in Eq. (2.57) is true for every function F and every operator A £ L(X,Y). Let x £ domip and x* £ detp(x). Then (see Theorem 2.4.2) cp(x) + ip*(x*) < (x,x*)+e. By Eq. (2.56) there exists y* £ Y* such that F(x,Ax) + F*(x* - A*y\y*) < (x,x* - A*y*) + (Ax,y*) + e, Formulas for conjugates and s-subdifferentials 123 i.e. (x* - A*y*,y*) =: (x*,y*) £ deF(x,Ax). Therefore x* = x* + A*y*, which proves that the inclusion "C" in Eq. (2.57) holds, too. • Note that (viii) V (hi) => (i) => (ii), (iv) V (v) V (vi) =>• (ii), (vii) => (iv) A (v) A (vi), and (ii) =$• (i) if X is a normed space. Note also that similar properties to those stated in the second part of Proposition 2.7.2 can be given for the situations of Theorem 2.8.1. Corollary 2.8.2 Under the conditions of Theorem 2.8.1 we have that inf F(x,Ax) = max ( - F*(-A*y*,y*)). (2.59) xeX y*€Y* Furthermore, x~ is minimum point for (i) there exist Xo £ R, B £ "Bx and VQ £ Ny0 such that V0CA{[f<\0\nB)-[g<\0\; (ii) for every U £ J^x there exist A > 0 and V £ Ny0 such that VcA([f AxQ; (iv) X,Y are metrizable, 0 £ lb(A(dom f) — domg), f and g have proper conjugates, either f, g are cs-closed and gr A is cs-complete, or f, g are cs- complete; 124 Convex analysis in locally convex spaces (v) X is a Frechet space, Y is metrizable, f, g are li-convex functions and Oeib (A(dom /) - dom g); (vi) X is a Frechet space, f,g are Isc and 0 £ lb(A(dom f) — domg); (vii) X, Y are Frechet spaces, f,g are Isc and 0 £ tc(^4(dom/) — domg); (viii) dimlo < oo and 0 € *(A(domf) — domg); (ix) g is quasi-continuous and A(dom/) and domg are united; (x) Y = Wl, qri(dom/) ^ 0 and 4(qri(dom/)) l~l Momg ^ 0. Then for every x* 6 X*, x 6 domy? and e > 0 we have: p'OO = min{/*(** - A*y*) + g*(y*) | y* € F*}, de Proo/. We apply Theorem 2.8.1 to A and F : X x Y -> 1 defined by F(x,y) := f(x) + g(y). It is easy to see that if one of the conditions (i)- (ix) holds, then the corresponding condition of Theorem 2.8.1 is verified. If (x) holds, using the properties of the algebraic relative interior in finite dimensional spaces (p. 3) and Proposition 1.2.7, we have *(i4(dom/)-domff) = i(A(dom/))-i(domg) = ^(qri(dom/))-i(domg), and so (viii) holds, too. The conclusion follows then applying Theorems 2.8.1, 2.3.1 (viii) and Corollary 2.4.5. • Note that, as in Theorem 2.8.1, we have that (viii) V (iii) => (i) => (ii), (iv) V (v) V (vi) =» (ii), (vii) =>• (iv) A (v) A (vi), (ii) => (i) if X is a normed space, and of course, as mentioned in the proof, (ix) =>• (viii). Applying the preceding result we obtain a formula for normal cones. Corollary 2.8.4 Let A 6 Z(X,Y) and L C X, M C Y be convex sets. Suppose that one of the following conditions is verified: (i) there exists XQ S L such that Ax0 6 int M, (ii) X,Y are of Frechet spaces, L,M are li-convex and 0 € lb(A(L) — M), (iii) dimF N(Lr\A-l(M);x) = N{L;x) + A*{N(M; Ax)). (2.61) Formulas for conjugates and e-subdifferentials 125 Proof. Using the preceding theorem for f := IL, 9 •= IM and A, formula (2.61) follows from formula (2.60). • Corollary 2.8.5 Under the conditions of Theorem 2.8.3 we have the fol lowing relation, called the Fenchel-Rockafellar duality formula, inf. {f(x)+g(Ax)) = ma* ( - /'(-AY) - g*(y*)). igA y €' Furthermore, x is a minimum point for f + go A if and only if there exists y* G Y* such that -A*y* £ df(x) andy* £ dg(Ax). Proof. We proceed as in Corollary 2.8.2 (or apply this corollary). • Two particular cases of Theorem 2.8.3 are important in applications: / = 0 and A = Idx- The next theorem is stated even for A replaced by a convex process C. Theorem 2.8.6 Let g £ A(F) and C : X =4 Y be a convex process. Assume that 0 G D, where D := ImC — domg. Consider YQ := linD and the function ip : X —> R, (i) for every U G Nx there exist A > 0 and V € Ky0 such that V C [g < A] - e(XU); (ii) there exist XQ € R, B € 3x and V0 € Ny0 such that V0 C [g < X] - G(B); (iii) there exists j/o € dorngfllmC such that g is continuous at yo; (iv) X, Y are metrizable, g has proper conjugate, either g is cs-closed and C is cs-complete, or g is cs-complete and C verifies (Hz), and 0 £ lbD; (v) X, Y are Frechet spaces, g, C are li-convex and 0 6 lbD, (vi) dim Y0 < oo and 0 6'D. Then Vi'el* : ip*{x*)=mm{g*{y*) \ x* G C*( A(x), and e > 0 then de(p(x) C C* {deg(y)) (with equality i/grC is a linear sub space). Proof. Let x* £ X*. Consider F : X x Y -)• I, F(x,y) := g(y) - (x,x*). Then If one of the conditions (i)-(vi) holds then the corresponding condition of Theorem 2.7.4 holds. (In fact, in case (iv), if g is cs-complete and C satisfies (Ha;) one verifies directly that the function $ from the proof of Theorem 2.7.4 satisfies (E(x,z)).) So, by Theorem 2.7.4, there exists y* £ Y* such that + sup{(a;,a;*) + (z,-y*) - igre(x,z) \ x € X,z € Y} = 9*(v*) + t-(gr e)+ (x*, -y") = g*{y") + tgr e* (y*,x*). Since V/eT : V*(x*) Theorem 2.8.7 Let /, VoC[/ (ii) for every U £ Nx there exist A > 0 and V £ N.*-,, S«C/J that VC[f (iii) there exists XQ G dom / n dom g such that g is continuous at XQ ; (iv) X is metrizable, f,g have proper conjugates, f is cs-closed, g is cs-complete and 0 G l6(dom/ - domg); (v) X is a Frechet space, f,g are li-convex and 0 G li>(dom/ — domg); (vi) X is a Frechet space, f,g are Isc and 0 € l6(dom/ — domg); (vii) X is a Frechet space, f,g are Isc and 0 G IC(dom/ — domg); (viii) dimX0 < oo and 0 G *(dom/ — domg); (ix) g is quasi-continuous and dom/ and domg are united; (x) X is a Frechet space, f,g are li-convex and (0,0) G lb[{(x,x) \ x G X} — dom / x dom g). Then for x* € X*, x G dom / Pi domg and e > 0 we have: (/ + de(f + 9)(x) = {J{deif(x) + dE2g(x) \ei,e2 > 0, ex + e2 =e), d(f + g)(x) = df(x) + dg(x). Proof. Taking A = Idx, the conclusion follows from Theorem 2.8.3 under conditions (i) - (iii), (v) - (ix). If (iv) holds, taking Y = X and $(x,y) := f(x) + g(x — y), condition (iv) of Theorem 2.7.1 is verified. If (x) holds consider F : X x X -)• I, F{x,y) := f{x) + g{y) and A : X -> X x X, A(x) :— (a;, a;). Then condition (v) of Theorem 2.8.6 holds; applying it we obtain the conclusion, taking into account that A*{x*,y*) = x* + y*. • The same implications as in Theorem 2.7.1 (mentioned in Proposition 2.7.2) hold. Corollary 2.8.8 Let X be a Frechet space and f, g G A(X). /// and g are li-convex then for every x G *6(dom/ + domg) we have (/Dg)(x) = (/*+$*)•(*) = m&x{(x,x*) - f*(x*) - g*(x*) \ x* € X*}. (2.64) Proof. Let a;o G s6(dom/ + domg) and consider h G A(X), h(x) := g(a;o — x). Then dom/i = XQ — domg, and so dom/ - dom/i = dom/ + domg — XQ- Therefore 0 G l6(dom/ — dom/i), and so condition (v) of 128 Convex analysis in locally convex spaces Theorem 2.8.7 is satisfied. From formula (2.63) applied for x* = 0 we get (fn9)(x0) = inf. (/Or) + h{x)) = -(/ + h)*(0) = - min (/•(**)+ >»*(-**))• X*£X * But h*{—x*) = sup{- (x,x*) — g(xo — x) \ x € X} = g*(x*) — (x0,x*), and so (2.64) holds for x = x0. O Of course, one can obtain the conclusion of the preceding corollary also for other situations corresponding to conditions of Theorem 2.8.7. For x sucn example, if there exist A0 £ M, B € "S>x, VQ € Nx0 ( ) that Vo C [/ < A0] n B + [g < A0], where X0 - aff(dom/ + dom^), then Eq. (2.64) holds. Proposition 2.8.9 Let f,g € A(X) and take D = dom/ - domg. If dim(lin£>) < oo then {D = rintZ?, while if X is metrizahle, f,g have proper conjugates, f is cs-closed, g is cs-complete and tbD ^ 0 then lbD = rint D; similarly for the situations corresponding to conditions (v)-(vii) of Theorem 2.8.7. Suppose that X is a Banach space and f, g are li-convex. Then for every x € lbD there exist n, A > 0 such that (x + nUx) n aff D C [f < A] n \UX - [g < A] D \UX- (2.65) Proof. The first part follows from Proposition 2.7.2 taking Y = X and $(x,y) =f(x)+g(x-y). Suppose that X is a Banach space and /, g are li-convex; consider x £ lbD. Replacing / by /, f(u) — f(u + x), we may suppose that x = 0. It follows that condition (v) of Theorem 2.8.7 holds, and, as noted after its proof, condition (i) is verified. Therefore there exist 77 > 0, Ao G B. and B £"Bx such that rjUx n aff D C [/ < Ao] n B - [g < A0]. Taking A' > max{Ao, 0} such that B C X'Ux and A = A' + n we obtain that rjUx n aff D C [/ < A0] 0 B - [g < A0] D (B + nUx) c [/ < A]n xux - [f < A]n \ux. The proof is complete. • Another important result is the following. Formulas for conjugates and E-subdifferentials 129 Theorem 2.8.10 Let Y be ordered by the convex cone Q, f £ A(X), H : X —> Y' be convex and g £ A(y) be Q-increasing on H(domH) + Q. Then tp :— f + goH is convex. Assume that 0 £ D, where D :— H(domHC\ dom/) — domg + Q, and consider Yo := linD. Assume that one of the following conditions holds: (i) for every U £ J^x there exist A > 0 and V € Ky0 such that V C H (XU n [/ < A] n domF) - [g < A] + Q; (ii) there exist XQ 6 R, B £ 1$x and VQ £ Ny0 such that V0 C H (B n [/ < A0] n domH) -[g< A0] + Q; (iii) there exists XQ £ dom/ fl if-1 (domg) such that g is continuous at H(x0); (iv) X, Y are metrizable, f, g have proper conjugates, either f, g are cs-closed and epiif is cs-complete, or f,g are cs-complete and epiH is cs-closed, and 0 G lbD; (v) X, Y are Frechet spaces, f,g, epiH are li-convex and 0 £ tbD; i (vi) dim Y0 < oo and 0£ D; (vii) g is quasi-continuous, and domg and H(domH f~l dom/) + Q are united. Then for x* £ X*, x £ dormp = dom/ n H~1(domg) and e >Q, we have: + deV{x) = |J {0Er (f + y*° H){x) \y*£Q n dE2g(H(x)), El + e2 = e} . (2.67) Proof. First observe that for all x* £ X*, y* £ Q+ and x £ dom<£ we have V'OO < (/ + V* o H)*(x*) + g*(y*), (2-68) + de f{x) + (y* o H)(x) + (f + y*o H)* (x*) > (x, x*), g(H(x))+g*(y*)>(H(x),y*), whence, adding them side by side, we get g*(y*) + (/ + y* ° H)* (x*) > (a;, a;*) - f{x) + (y* o H){x) + (f + y*o H)* {x*) < (x, x*) + ex, g(H(x))+g*(y*)<(H(x),y*)+e2. Adding them side by side we get ip(x) + g*(y*) + (/ + y* ° H)* (x*) < {x,x*) + e. Using Eq. (2.68) we obtain that x* € deip(x). Therefore Eq. (2.69) holds. Let F(x, y) := f(x) + g(y) and G : X =t Y with gr C := epi H; then F £ A(X x Y) and 6 is convex. Since g is increasing on H(dom H) + Q, it follows that (p(x) = int{F(x,y) + iep\H{x-,y) \ y 6 Y] for every x € X. Hence ip is the marginal function associated to the convex function F + iepi H ', hence ip is convex. If one of the conditions (i) — (vi) is verified, then F and 6 satisfy the corresponding condition of Theorem 2.7.4. (When / and g are cs-complete and have proper conjugates, similarly to the proof of Proposition 2.2.17, we get that F is cs-complete.) As noticed after the proof of that theorem, also the perturbed function F, F(x,y) = F(x,y) — {x,x*), satisfies the same condition. Therefore there exists z* £ Y* such that a:= nf , j .1TUix)+g{y)-{x,x'')) = inf (f(x)+g(y)-(x,x*) + (z,z*))=:(3. {x,y+z)EepiH Using again the fact that g is increasing on H(domH) + Q, we have a = nf J w j?^ n VW + 9{y) ~ (x,x*)) xedom H yeH(x)+Q = inf (f(x)+g(H(x)) - {x,x*)) = ~ P = inf inf (f(x) + g(y) - (x, x*) + (H(x) +q-y, z")). iGdomif q€Q, y€Y It follows that P = -oo if z* $ Q+. If z* £ Q+ then -0= sup sup((x,x*)-f(x)-(H(x),z*) + (y,z*)-g(y)) x€dom Hy£Y = (/ + Z*o #)*(*•) + • K, F(x,y) := f(x)+g(H(x)+y)- (x,x*). Because 0€ D and g is Q-increasing on H(domH) + Q, there exists 1 xo € dom/ n H~ (domg). It follows that F(x0, •) is quasi-continuous and {0} and D — Pry (dom F) are united. Applying Theorem 2.7.1(ix) we have that infxex F(x,0) = maxy.ey.(—F*(0,y*)). With a similar calculation as above we obtain that Eq. (2.66) holds. + Let x 6 donup and x* 6 d£ip(x). Using Eq. (2.66), there exists y* € Q such that ip*(x*) = (f + y* o H)*(x*) + g*{y*). It follows that {f+y*oH){x)+{f+y*oHY{x*)-(x,x*)+9{H{x))+g*{y*)-{H{x),y*) < e. Taking £l := (/ + y* o tf)(x) + (/ + y* o #)*(x*) - (x,x*) (> 0 by the Young-Fenchel inequality) and e2 = e - ei, we have that y* € d£2g(H(x)) and Z* € d£l(f + y* oH)(x). Therefore the inclusion C holds in Eq. (2.67); taking into account Eq. (2.69), we have the desired equality. D We use the preceding theorem to obtain formulas for conjugates and subdifferentials of functions of "max" type. Corollary 2.8.11 Let /i, ...,/„ E \(X) and y.X^W, ¥>(*):=/i(z)V---V/„(z). Suppose that dom ip = f}™=1 dom /i ^ 0 and e € IR+. For x € dom ip denote by I(x) the set {i G l,n | fi(x) = >p*{x*) = min{(Ai/i + • • • + An/n)*(x*) | (Ai,..., An) e A„}, 132 Convex analysis in locally convex spaces while for every x £ dom n fn){x) (Ai, . . . , A„) G A„, Xi x 77 e [0,e], Yli=1 ^ ^ - ¥>(x)+ri-£o} , 5 AJi {x) (Al! An) € A dV{x) = U { (E"=1 ) I ' • •' "' Vifl{x) : Ai = o}. Proof. Let us consider the functions H:X->(W\Rl), H(x):=( (A(*)>••"'/«(*)) if x € fX=i dom/,, "•" [00 otherwise, 3: «"-•«, g(y):=yiV.---Vyn. It is clear that condition (iii) of the preceding theorem is verified. Taking into account the expressions of g* and deg(y) given in Corollary 2.4.17 we obtain immediately the relations from our statement. D An application of the previous result is given in the following example. Example 2.8.1 Let / € A(X) and consider /+ := / V 0. Then f df(x) if f(x) > 0, df+(x) = l U{W)(x)|Ae[0,l]} if f(x) = 0, { d(0f)(x) = didomf(x) if f(x)<0. A useful particular case of the result in Corollary 2.8.11, in which we can give explicit formulas for ip* and detp without supplementary conditions, is presented in the following corollary. Corollary 2.8.12 Let f{ 6 A(X,) for i € T~n and For x = (xi,... ,xn) S domtp = Yl7=i dom/j consider I(x) := {i G l,n | fi(Xi) = For x — (xi,..., xn) E dom n dMx) = {j{~[[ .=1dei(^fiKxi)\ (Ai,...,A„)GAn, et > 0, n V~^n 1 E £i = £ Xi Xi £ i=o ' Z^i=1 ^ ^ - ^ - o) > S^^-lJln^^Ai/OC^)! (A1,...,An)€A„, VigJ(a:) : A; = o} . Proof. We apply the preceding corollary to the functions fa : X ->• R, /i(a;) := fi(xi), then we use Theorem 2.3.1 (viii) for arbitrary n and Corol lary 2.4.5. • Other situations when one has explicit formulas for tp* and d 3 Proof. When / and g verify one of the conditions (i)-(iii), (v), (viii)— (x) of Theorem 2.8.7 and A,/x > 0, then the functions A/ and fig also verify the same condition. If condition (vi) or (vii) holds then, by the relations among the classes of convex functions on page 68, condition (v) holds. So, (Xf + ng)*{x*)=Toia{(Xf)*{u*) + (jjig)*{v*) \u*+v* =x*}and d(\f + ng)(x) = d(Xf)(x) + d(fig)(x). Using Corollary 2.8.11 one obtains the conclusion. • Taking into account that for x G dom /, one has df(x) + N(dom f, x) = df(x) (see also Exercise 2.23), d(Xf)(x) = \df(x) for A > 0 and d{0f)(x) = N(domf,x), we get for f,g G A(X) and x G X with f(x) = g(x) G R, U{W)W+8WWI(A,|f)6A2} -co(df(x)Udg{x)) + N(domf,x) +N(domg,x). This formula can be extended easily to a finite number of functions. Using Corollary 2.8.12 we get the following result concerning the max- convolution. 134 Convex analysis in locally convex spaces Corollary 2.8.14 Let /i, /2 € A{X) and e £ 1+ . For every x* £ X* we have that (hQf2y(x*)=rnin{(\1f1y(x*) + (\2f2y(x*) | (A1;A2) £ A2}. (2.71) Suppose that (/i0/2)(a:) = max{/i(2;i),/2(a;2)}, where xi £ dom/i, x~2 £ dom /2 and x = x~i + x2. Then de{hOh){x) = \J{dEl(^h)(xi)ndE2(\2f2)(x2) I (Ai,A2) e A2, ei,e2>0, £i+e2 Furthermore, if fi(x\) = f2(x2) *^en Eq. (2.73) is verified, while if fi(xi) > 72(^2) *^en i?g. (2.74) is verified, where d(fi0f2)(x)=[j{d(X1f1)(x1)nd(X2f2)(x2) I (A1;A2) G A2}, (2.73) d(fi0f2)(x) = dMxx) nN(domf2-:x2). (2.74) Proof. Let x* £ X*; then {fi0f2)*(x*) = supieX((x,x*) -inff/^) V/2(z2) I *! + x2 = x}) = sup{(ii,a;*) + (a;2,a;*) - fi(xi) V f2(x2) \x1} x2 £ X) = min{(A1/1)*(x*) + (A2/2)*(s*) | (Ai,A2) £ A2}, the last equality being obtained by using Eq. (2.70). The inclusion "D" of Eq. (2.72) can be verified easily. Consider x* £ de(fi0f2)(x). By Eq. (2.71), there exist (Ai, A2) £ A2 such that (/i0/2)*(x*) = (Ai/i)*0O + (A2/2)*(x'). (2.75) Using the preceding relation we obtain that 0<[(A1/1)(^1) + (A1/1)*(2;*)-(5;1,a;*)] + [(A2/2)(i2) + (A2/2)*(:c')- < (fiOh)(x) + (/iO/2)*00 - (x,x*) < e. Taking e, := (Aj/i)(zj) + {Xifi)*(x*) - (x~i,x*) > 0, i £ {1,2}, using the preceding relation and Eq. (2.75), we get (/i0/2)(z) +£1 - Ai/i(3fi) +e2 - A2/2(x2) < e, and so x* belongs to the set on the right-hand side of relation (2.72). Formulas for conjugates and s-subdifferentials 135 Let us prove now the equalities (2.73) and (2.74). The inclusions "D" follow directly from Eq. (2.72). Let us prove the converse inclusions. Let x* G d{faOfa)(x) = d0(faOfa)(x). By Eq. (2.72), there exist (Ai, A2) G A2 and £i, £2 > 0 such that x* £dei(Xifi)(x1)ndea(Xif2)(x2), £1 + e2 < Xifi(xi) + A2/2(x2) - (faOfa){x) < 0. Therefore £1 = £2 = 0 and Ai/i(xi) + A2/2(x2) = {fa(>fa){x); hence x* belongs to the set on the right-hand side of Eq. (2.73) when /i(xi) = fa{x2)- If fi{x~i) > /2(^2), the preceding relation shows that A2 =0, Ai = 1, and so x* G 9/i(11), x* G 0(0 • /2)(z2) = <9idom/2(^2) = -/V(dom/2;x2). This shows that x* belongs to the set on the right-hand side of relation (2.74). • Note that in Eq. (2.72) we can take E\ + £2 = £ + •••. Moreover, if /1 is continuous at x\ and fa is continuous at a;2, then in Eq. (2.74) we have d(fiOfa)(x) = {0}. Indeed, in this situation, N(domfa;x~2) = {0} (since x~2 G int(dom/2)); since /1O/2 is continuous at x, d(fiQfa)(x) ^ 0. In particular, it follows that X\ is a minimum point of fa. Furthermore, in this situation, /1O/2 is even Gateaux differentiate. Introducing the convention that 0 • df(x) := N(domf;x) for / G A(X) and x G dom/, formula (2.75) may be written in the form d(fa0fa)(x) = dfa(x1)odfa(x2), where A o B represents the harmonic sum of the sets A and B, which generalizes the inverse sum of A and B. In order to have more explicit formulas in Corollary 2.8.11 it is necessary to impose some supplementary conditions. Let us give such conditions and the corresponding formulas in three situations for n = 2. Corollary 2.8.15 Let fa, fa G A(X), e G 1+ and E, >p := max{/i,/2}. Suppose that one of following conditions is verified: (i) there exists XQ G dorafa n dom/2 such that fa is continuous at XQ; (ii) X is a Frechet space, fa, fa are li-convex and 0 G l6(dom/i — dom/2); (iii) dim X < 00 and ^dom fa) n *(dom fa) ^ 0. 136 Convex analysis in locally convex spaces Then, for all x* € X* and x £ dom/i D dom/2 we have: ¥>*(*•) = min{(Ai/i)*(a;I) + (A2/2)*(^) | (Ai, A2) e A2, x\ + x*2 = x*}, dMx) = \J{dEl(Xifi)(x)+dE2(X2f2){x) I (Ai,A2) G A2, e0>ei,e2 > 0, e0+ei + e2 = e, Xifi(x) + X2f2{x) > dtp(x) = \J{d(X1h)(x)+d(X2f2)(x) I (Ai,A2) G A2> Xih(x) + x2f2{x) = tp(x)}. a Let * £ X* and consider the functions /i,/2 : X —>• K defined by /i(a;) :=max{(i,a;;),...,(j;,<)}, /2(x) := |{a;,a;*)|. Applying the preceding corollary, we get the following formulas for every xeX: d x A h( ) = { ]C"=1 ^i I (Ai, • • •, A„) € A„, ^ = 0 if (x, x*) < /!(a;)} , r {x*} if (x,x*) >0, 0/2 (*) = < {Az* I A € [-1, +1]} if (x, x*) = 0, 1 {-a;*} if (x,x*) <0. 2.9 Convex Optimization with Constraints Let us come back to the general problem of convex programming as con sidered in Section 2.5, (P) min f(x), x £ C, where / € A(X), C C X is a convex set and C (~l dom/ ^ 0; the spaces considered in the present section are separated locally convex spaces if not stated explicitly otherwise. Since x is solution of (P) exactly when it minimizes f + ic, x is solution of (P) if and only if 0 € d(f + i Theorem 2.9.1 (Pshenichnyi-Rockafellar) Let f € A(X) and C C X be a convex set. Suppose that either dom / n int C ^ %, or there exists xo € dom/ (~1 C, where f is continuous. In these conditions x G C is a solution of (P) if and only if df{x) l~l {-N(C;x)) ^ 0. Convex optimization with constraints 137 Proof. In the conditions of our statement we have that VxeCndom/ : d(f + ic){x) = df(x) + duc{x) = df(x) + N{C;x), whence the conclusion is obvious. • Very often the set C from problem (P) is introduced as the set of solu tions of a system of equalities and/or inequalities. Let Y be ordered by a closed convex cone Q CY and H : X -» Y' be a Q-convex operator; the set C := {x € X \ H(x) (P0) min f{x), H{x) (Py) min f(x), H(x) /(l) if $:IXF4l, *(*,y):=«(x, y) := {| ^^^ (2.76) v 'wy ' oo otherwise. v ' The problem (Py) becomes now (Py) min $(x,y), i£l We obtain, without difficulty, that $ is convex. Moreover $*(z*,-y*) = sup{(x,a;*) + (y, -y*) - *(x,y) \x€X,yeY} = sup{(x,x*) - (y,y*> - /(z) | x € X,y £ Y,#(aO Hence $*(x*,-y*) = sup{(x,x*) - f(x) - (H(x),y*) \x£domH} if y* e Q+ and $*{x*,-y*) = oo if y* g Q+. 138 Convex analysis in locally convex spaces The function ) + (H(x),y*) ifxGdomtf, L:XxQ+->R, L(x,y*):= { ^x ^ oo if x $. dom H, is called the Lagrange function (or Lagrangian) associated to problem (Po)- The above definition of L shows that L(x,y*) — f(x) + (y* o H){x) with the convention that y*(oo) = oo for y* G Q+, convention which is used in the sequel. By what was shown above, we have that Vy*eQ+ : &(0,-y*) = 8up(-L(x,yt)) = -hdL(x,y*). xex x^x Therefore, the dual problem of problem (Po) (see Section 2.6) is (£>o) max ( - $*(0, (Y',Q) be a Q-convex operator, where Q (S) 3x0edomf : -H{x0) € intQ. Then the problem (D0) has optimal solutions andv(Po) = v(Do), i.e. there exists y* G Q+ such that inf{/(x) | H(x) O£d{f + y*o H)(x) and (H(x),y*) = 0; (hi) there exists y* G Q+ such that (x, y*) is a saddle point for L, i.e. Vx E X, Vy* G Q+ : L(x,y*) < L{x,y*) < L{x,y*)- (2.77) Proof. The Slater condition (S) ensures that (XQ, 0) G dom $ and $(x0, •) is continuous at 0, where $ is the function denned by Eq. (2.76). Applying Theorem 2.7.1, there exists y* G Y* such that v(P0) = -$*(0,-y*). If v(Po) = —oo, then $*(0,2/*) = oo for every y* G Y*; in this case we may Convex optimization with constraints 139 take y* = 0. If f(Po) > —oo, using the expression of $*, we have that y* e Q+. Therefore v(P0) = inf{/(x) | H(x) = v(D0). (i) =S- (ii) Of course, x being a solution for (P0), we have that H(x) f(x)=v(P0)=in{{L(x,y*)\xeX}, whence Vx € X : f(x) + (H(x),?) < f(x) < f(x) + (H(x),y*)- (2.78) Relation (2.78), without the term from its middle, says that x is a minimum point for f + y* o H, and so 0 € d(f + y* o H)(x); taking x = x in relation (2.78) we obtain that (H{x),y*) - 0. (ii) =>• (hi) Since 0 € d(f + y* o H)(x) we have Vx G X : L(x,r) = /(x) + (^(x),y*) < f{x) + (H(x),y*) = LfoiT). Furthermore, for y* £ Q+ we have that L(x,y*) = f{x) + (H(x),y*) < f(x) = f{x) + (H{x),y*) = L(x,?). Therefore Eq. (2.77) holds, i.e. (x,y*) is a saddle point of L. (iii) => (i) Taking successively y* = 0 and y* = 2y* on the left-hand side of Eq. (2.77) we obtain that (Hix),y*) = 0. Using again the left-hand side of Eq. (2.77), we obtain that (i?(x), y*) < 0 for every y* e Q+; thus, using the bipolar theorem (Theorem 1.1.9), we have that —i?(x) € Q++ = Q, i.e. x is an admissible solution of (P0). From the right-hand side of relation (2.77) we get Vx e X, H{x) Vxedom/ : dif + y*oH)ix)=8fix)+diy*oH)ix); 140 Convex analysis in locally convex spaces the fact that Q is closed was used only for the implication (iii) =4- (i) of the above theorem, to prove that x is an admissible solution. An important particular case is when there are a finite number of con straints. Let /, gi,... ,gn 6 A(X) and consider the problem (Pi) min f(x), gi{x) < 0,1 < i < n. The dual problem of (Pi) is (Di) max mix€X(f(x) + \igi(x)-] hAn(/n(i)), Ai > 0,..., A„ > 0. Then the following result holds. Theorem 2.9.3 Let f,gi,...,gn € A(X). Suppose that the Slater con dition holds, i.e. 3xo € dom/, \/i£l,n : gi(x0) < 0. Then: (i) the dual problem (Di) has optimal solutions and u(Pi) = v(D{), i.e. there exist (Lagrange multipliers) Ai,..., A„ £ M+ such that mf{f(x)\gi(x)<0,...,gn(x)<0} -ini {f(x)+J1g1(x)-\ + Xn9n(x) \ x 6 X) . (ii) Let x 6 dom /; x is a solution of problem (Pi) if and only if gi(x) < 0 for every i6l,n and there exist Ai,..., An € K+ such that Xigi(x) = 0 for i £ l,n and 0 € d (/ + Aiffi + ... + *ngn) \XJ- If the functions g\,..., gn are continuous at x, the last condition is equiv alent to - 0 € df(x) + A !0ffi(2c) + ... + \ndgn(x). n Proof. Let us consider H : X -> (M *,E!J:), H(x) := (gi(x),.. .,gn(x)) for x G nr=idom5j, i?(x) = oo otherwise; H is R™-convex. The result stated in the theorem is an immediate consequence of the preceding theo rem. When gi are continuous at x one has, by Theorem 2.8.7 (iii), that d(f + Aiffi + ... + \n9n)(x) = df(x) + d(Xigi)(x) + ... + d(\ngn)(x), and d(\igi)(x) = Jidg^x). D Convex optimization with constraints 141 Corollary 2.9.4 Let f,gi,--.,gn '• X —> R be Gateaux differentiable continuous convex functions. Suppose that there exists x$ £ X such that 9i{xo) < 0 for every i 6 l,n. Then x £ X is solution of problem (Pi) if and only if gi(x) < 0 for every i £ l,n and there exist Ai,..., An £ R+ such that —\7f(x)=\iVgi(x) + ... + \nVgn(x) and \igi(x) = 0 for i £ l,n. • Corollary 2.9.5 Let g 6 A(X), 7 £ ] inf g, 00 [ and S£ [5 < 7]. Tften N([g < l\,x) = \J {d(Xg)(x) | A > 0, X(g(x) - 7) = 0}. (2.79) Proof. The inclusion "D" in Eq. (2.79) holds without any condition on g and 7. Indeed, let x* £ d(Xg)(x) for some A > 0 with X(g(x)— 7) = 0. Then for x £ [g < 7] we have that (a: — x, x*) < Xg(x) — Xgix) = X(g(x) — 7) < 0, and so x* £ N([g < ^],x). Let x* £ N([g < j];x). Then x is a solution of the problem (Pi) min {x, —x*), h(x) := g(x) — 7 < 0, whence, by Theorem 2.9.3, there exists A > 0 such that Xh(x) = X(g(x) — 7) = 0 and 0 G d(-x* + Xh){x), i.e. x* £ d{Xh){x) = d(Xg)(x). Therefore the inclusion "c" of Eq. (2.79) holds, too. • Note that d(Xg)(x) = Xdg(x) if A > 0 and d(0g)(x) = didomg(x) = N(domg;x). In the case of normed vector spaces, taking A = X in Proposition 3.10.16 from Section 3.10, we have another sufficient condition for the validity of formula (2.79). Note that we can obtain the characterization of optimal solutions in Theorem 2.9.3, when the functions gi are continuous, using Corollary 2.9.5 and the formula for the subdifferential of a sum. Indeed, x £ dom / is a solution of (Pi) if and only if x is minimum point of the function / + icx + htc„, where Cj := {x | gi(x) < 0}. By hypothesis dom/fl f)"=1 int d ^ 0, and so, for every x £ dom/ (~l Hili^i we have d(f + tCl+... + icn) (x) = df{x) + N(Cf,x) + ... + N(Cn;x). Using formula (2.79) we obtain the desired characterization. Theorem 2.9.2 can be extended further to the case when there are also linear constraints. Let us consider the problem (P2) min f(x), H{x) L2 : X x (Q+ x Z*) -• 1, L2(aM,V) := /(a:) + (H(x),y*) + (Tx,z*). The dual problem associated to (P2) is (£>2) max (mixeXL2(x,y*,z*)), y* G Q+, z* G Z*. Then the following result holds. Theorem 2.9.6 Let f G A{X), H : X -> (Y,Q) be a Q-convex oper ator, where Q C Y is a closed convex cone, T G L(X,Z) be a relatively open operator and x G dom/. Suppose that there exists XQ € dom/ such that f,H are continuous at xo, —H{XQ) G int<5 and TXQ = 0. Then the problem (D2) has optimal solutions and V{PQ,) = v{D2), i.e. there exists y* G Q+, z* G Z* such that inf{/(x) I H(x) Furthermore, the following statements are equivalent: (i) x is solution of problem (P2); (ii) H(x) T*z* G df(x) + d(y* o H)(x) and (H(x),y*) = 0; (iii) there exists (y*,z*) G Q+ x Z* such that (x,(y*,z*)) is a saddle point of L2, i.e. L2(x,y*,z*) < L2{x,y*X) < L2{x,y*,T) for all x G X and all y* G Q+, z* G Z*. Proof. Let us consider the perturbation function f(x) if H(x) < y, T(x) = z, *:Ix(7xZ)->I, $(1,y,z) := , Q v ; ' ^ 'y' y ' 00 otherwise. We intend to prove that condition (i) of Theorem 2.7.1 is verified. Note first that PryxZ(dom$) = {{H{x) + q,Tx) | x G dom/, q £ Q} C Y x ImT. A minimax theorem 143 Taking into account that / is continuous at xo G dom/, there exists Uo G Nx(io) such that Vxe% : f{x) Since —H{XQ) G intQ, there exists Vo G Ny such that —H(XQ) +VQCQ. There exists V € Ny such that V + V CV0. Since if is continuous at x0, there exists {/ € Nx(zo)) [/ C t/o, such that H(x) € H(x0) — V for every x € U. Since T is relatively open, there exists W G NimT such that W C T(LT). Of course, V x W G Nyximr(0,0); let (j/,z) G V x W. There exists x £U such that Tx = z. Since x G f/, we have that i?(i) = H(xo) — y' for some j/' G V. Then y - H(x) =y + y'- H(x0) G -#(x0) + V + V C -#(x0) + V0 C Q; hence -ff(x) Y* x Z* such that v(P2) = -4>*(0, -y*, -z*). But $*(0, -y*, -z") = sup(w)6XxyxZ((i/, -y*) + (z, -z*) - $(x,y, z)) = sup{-(y,y*) - (z,z*) - f(x) | H(i) Thus **cn _„* —?*\ - J -infiex^Ci.y*,^*) if 2/* e Q+, * lU' y ' z) ~ \ oo if j/* £ Q+. + Therefore y* G Q if u(P2) G K and we can take j7* = 0 if v(P2) = -oo. The proof of the second part is completely similar to that of Theorem 2.9.2. We note only that, / and H being continuous at xo, we have d(f + y* o H)(x) = df(x) + d(y* o H)(x) for every x G dom/. • 2.10 A Minimax Theorem In the preceding section we have obtained some results which are stated using saddle points. In the sequel we establish a quite general result on the 144 Convex analysis in locally convex spaces existence of saddle points as well as a minimax theorem. Let A and B be two nonempty sets and / : A x B -> JR. It is obvious that sup inf f(x,y) < inf sup f(x,y). (2.80) xeAvtB y£BxeA The results which ensure equality in the preceding inequality are called "minimax theorems," the common value being called saddle value. Note that if / has a saddle point, i.e. there exists (x,y) € A x B such that VxeA,Vy&B : f(x,y) < f(x,y) < f(x,y), (2.81) then max inf f(x,y) = minsup f{x,y), (2.82) x£A yeB yeB xeA where max (min) means, as usual, an attained supremum (infimum). Indeed, let (x,y) € Ax B satisfy Eq. (2.81). It follows that : iniB: sup f(x,y) < sup f(x,y) < f(x,y) < infB (x,y) < sup infB f{x,y). v£ x€A xeA y£ xeAv& Using Eq. (2.80) we obtain that all the terms are equal in the preceding relation, and so (2.82) holds (the maximum being attained at x and the minimum at y). Conversely, if Eq. (2.82) holds then / has saddle points. Indeed, let x G A and y e B be such that inf (x,y) = sup ini' f(x,y) = irtf swp f(x,y) = sup f(x,y). y£B xeAVZB y€Bx€A x£A Since infy6s(x,y) < f(x,y) < supx€Af(x,y), from the above relation it follows that all the terms are equal in these inequalities; this shows that (x, y) is a saddle point of /. So we have obtained the following result. Theorem 2.10.1 Let A and B be two nonempty sets and f : A x B ->• R. Then f has saddle points if and only if condition (2.82) holds. • The following result is an enough general minimax theorem (frequently utilized) which gives also a sufficient condition for the existence of saddle points. Theorem 2.10.2 Let X be a locally convex space, Y be a linear space, Ac X be a nonempty convex compact set and B CY be a nonempty convex set. Let also f : A x B -)• M. be a function with the property that f(-,y) A minimax theorem 145 is concave and upper semicontinuous for every y G B and f(x, •) is convex for every x E A. Then max inf f(x,y) — inf ma,xf(x,y). (2.83) X&A y£B y£B xEA If moreover Y is a locally convex space, B is compact and f(x, •) is lower semicontinuous for every x € A, then maxmmf(x,y) = min max f(x,y); (2.84) xeA y£B yeB x£A in particular f has saddle points. Proof. Let a e K be such that a > maxx£j4 infyeB f(x,y). For every x S A there exists yx 6 B such that f(x,yx) < a. Since f(-,yx) is upper semicontinuous at x, there exists an open neighborhood Vx of x such that f(u,yx) < a for every u G Vx. Since A is compact and A C UxeA ^> there exist x\,..., xp € A such that A C U?=i Vxt • Let 2/j := yXi • Consider the sets d :=co{{f(x,yi),...,f(x,yp))\xeA}cW, C2 := {(ui,...,up) \v,i>a\/i€ l,p} . It is obvious that C\ and Ci are nonempty convex sets, and C2 has non empty interior. Moreover C\ fl Ci = 0. In the contrary case there exist q 6 N, (Ai,..., A?) 6 Aq and Xi,..., xq € A such that Viel.p : a < Zf := 2^, >^jf{xj,yi). Since f(-,yi) is concave, taking xo = 13?=i ^j2^' we nave that io£i and Vi G l,p : a < Zj < f(x0,yi). There exists io £ l,p such that £0 £ VXi . Therefore f{xo,yi0) < a, a contradiction. Applying Theorem 1.1.3 there exists (/zi,..., fj,p) € W \ {0} such that . ,mf(x,yi) < V UiUi. t=l *—'i=l Letting u; -> 00 we have that /ij > 0 for every i, and so we can suppose that (/xi,..., Up) € Ap. Taking u = (a,..., a) we obtain that ^xeA : > . Uif{x,yi) < a. 146 Convex analysis in locally convex spaces Let 2/0 := Y%=i ViVi € B. Since f(x, •) is convex, we obtain that f(x, y0) < a for every x € A. Therefore a > infj/eBmaxxg^/(a;,2/), and so (2.83) holds. In Eq. (2.83) the suprema with respect to x are attained because the functions f{-,y), y £ B, and infyes f(-,y) are upper semicontinuous and A is compact. When B is compact and the functions f(x,-) are lsc for all x € A we obtain that the infima with respect to y € B are attained in Eq. (2.83), i.e. Eq. (2.84) holds. • Note that the convexity assumptions in the preceding theorem can be weakened. More precisely, we can suppose that A is a nonempty compact subset of a topological space, B is nonempty, and / satisfies the following two conditions (similar to concavity and convexity, respectively): Vxi,...,xp e A, V(Ai,...,Ap) e Ap, 3x0 &A,\fyeB : f(xo,y) > y\ ,Kf(zi,y), •'—'1=1 \/yi,...,yq GB, V(//!,...,/*,) € A,, 3y0 eB,Vx£A : v—yQ f(x,yo) < 2^,.=iHjf{x,yj). 2.11 Exercises Exercise 2.1 Let X be a linear space, / € A.(X) and x,u £ X. Prove that the mapping %p :]0, oo[—• R defined by ip(t) :— t • f(x + t~lu) is convex. Exercise 2.2 (a) Let I C R be an interval and /:/—>• R. Suppose that / is locally nondecreasing, i.e. for every a € I there exists e > 0 such that the restriction of / to IC\[a — s, a + e] is nondecreasing. Prove that / is nondecreasing. (b) Let g : [a, b] —• R (a, 6 € R, a < b) be a continuous function. 1) Assume that g'+{x) g R exists for every x € [a,b[; show that there exists x' £ [a,b[ such that g(b) — g(a) < g'+(x')(b — a). 2) Assume that g'-(x) € R exists for every x S ]a, 6]; show that there exists x" € ]a, b] such that g(b) — g(a) > gL (x")(b — a). (c) Let X be a locally convex space and A C X be an open convex set. If / : A —> R is locally convex, i.e. for every a G A there exists a neighborhood V of a such that f\vnA is convex, prove that / is convex. Exercise 2.3 Let / : Rn -*• R be a quasi-convex function and i^f. For Exercises 147 n n x eR consider the function fx : R ->• R, fx(t) = f(x + tx). Prove that / is lsc at ittViel" : fx is lsc at 0, / is use at JoVieR" : fx, is use at 0. If Rn is replaced by an infinite dimensional normed space then the above prop erties do not hold, generally. Exercise 2.4 Let X be a linear space, / : X —> R be a convex function and A > infiex fix)- Prove that {x G X | f(x) < \y = {x G X | f{x) < A}. Moreover, if X is a topological vector space and / is continuous, then int{x € X | f(x) < A} = {x G X | /(z) < A}. Exercise 2.5 Let X be a topological vector space and / : X —> R a convex function. Prove that: (a) [/<*]= cl[/ < *] f°r every t G ] inf /, oo[ if and only if / is lsc; (b) [/<*]= int[/ < *] for every t G ] inf /, cx)[ if and only if / is continuous on dom/; (c) [/ = *] = bd[/ < t] for every t G ] inf /, oo[ if and only if / is continuous (on X). Exercise 2.6 Let / : R" —)• R be a proper convex function for which all the partial derivatives df/dxi exists at a G int(dom/). Prove that / is Gateaux differentiable at a (even Frechet differentiate because / is Lipschitz on a neigh borhood of a). p Exercise 2.7 Let p G [1, oo[ and /:R2^I, f(x,y):={ (^tan|)^V^T^2 for x, y > 0, [ oo otherwise, where, by convention, arctan | := ^ for a; > 0. Prove that / is a lsc convex function, but not strictly convex. Exercise 2.9 Consider the function 1 /: c[o, i] -> R, fix) := /o yrrwoF^- 148 Convex analysis in locally convex spaces Prove that / is convex and Frechet differentiable of order 2 on C[0,1] with 1 2 dt V/(x)(«) = f -£L= dt, V f(X)(u, V) = ^ ,7/^3-2 Jo Vl+x2 Jo (l+2:2)vl + z for all u,v £ C[0,1]. Moreover, V2/ is continuous on C[0,1]. Exercise 2.10 Consider the function 1 ! /:L (0,1)->R, /(i):=/0Vl + (i(t)) *, where L1(0,1) is the Banach space of (classes of) Lebesgue integrable functions on the interval [0,1]. Prove that / is a Gateaux differentiable convex function with xu Vx.uSL^O.l) : V/(x)(u) = / :dt, Jo %/rn but / is nowhere Frechet differentiable. Exercise 2.11 Using properties of convex functions, prove that Va,6eR+, Vp,ge]l,oo[, £ + ^ = 1 : ab < ±ap + \b\ the equality being valid if and only if ap = bq, and Vn€ N, Vxi,...,x„ 6R+, Vai,...,a„ 6]0,1[, ai + ... + a„ = 1 : 1 n am + ... +anx„ > X™ • • -xZ , the equality being valid if and only if x\ = • • • = xn. In particular, VneN, Vsi,...,x„ €R+ : £(xi + ... + xn) > yX\---xn. Exercise 2.12 Let / : X —>• R be a proper function. Prove that / is convex if and only if / + x* is quasi-convex for every x* G X*. Exercise 2.13 Let n Exercise 2.14 Consider / : R ->• R, f(xi,...,x„) := In (X£=1 expxk). Prove that / is convex. Exercise 2.15 Consider p € R \ {0} and the function t" xP"->R, f{t,x):= nr=i*i' Prove that / is strictly convex for p €] — oo,0[U]n+l, oo[ and convex if p = n + 1. Exercises 149 Let c G Rn and A:={i£P" \(x\c)> 0}. Prove that the function (x\c)" g : A -> R, p(x) niu^ is strictly convex for p > n + 1 (it is possible to prove that g is strictly convex for p > n). The function g has been used by Karmarkar for establishing his interior point algorithm. Finally prove that the function h : Pn ->• R, h(x) := (T[n xA is strictly convex. Exercise 2.16 Let tp : R+ —> R+ be a continuous convex function such that ^i(t) = 0 •«• t = 0. Show that x = Jo i>'- (V>- «)) 7o V-H^-H*)) for every a > 0. Exercise 2.17 Let X, Y be normed spaces and T G L(X,Y). Prove that the function f:Y^R, f(y):=M{\\x\\\Tx = y}, is a sublinear functional. Moreover, if T is an open operator, then dom f = X and / is continuous. Exercise 2.18 Let / : Rk —> R be a strongly coercive proper lsc function. Prove that co(epi/) is a closed set. Exercise 2.19 Let X be a locally convex space, / G T(X), g : X —> R be a proper function and a, /3 > 0. Prove that co (co(/ + ag) + jig) = co (/ + (a 4- /3)g) and ((/ + ag)** + fig)" = (f + (a + fig)* . Exercise 2.20 Let X be a separated locally convex space and A,B,CcXbe nonempty sets. If C is bounded and A + C C B + C prove that A C coB. Exercise 2.21 Let (X, \\-\\) be a normed space, / G T(X) and L > 0. Prove the equivalence of the following statements: (i) dom/ = XandVx,t/GX : \f(x) - f(y)\ < L \\x - y||; (ii) 3a GR, Vi€X : /(x) < L ||x|| + a; (iii) VMGX : /oo(u) Exercise 2.23 Let X, Y be separated locally convex spaces, / € r(X), F € F(X x Y), x € dom / and A £ R, e £ R+ be such that [/ < A] and d£f(x) are non empty. Prove that: [/ < A]oo = [foo < 0], (ae/(x))00 = N{domf;x), (/*)«, = Sdom/, foe = Sdom/* and {v* £ y* | (F*)oo(0,t;*) < 0} = (Pry(domF)) . In particular, if {v* £ Y* | (F*)oo(0,u*) < 0} is a linear subspace then {0} and Pry (dom F) are united. Exercise 2.24 Let X be a separated locally convex space and / £ I\X). Consider K := {u £ X | /«,(w) < 0} and X0 := K D (-.K"). Prove that: (i) if is a closed convex cone, Xo is a closed linear subspace and f(x + u) = f(x) for all a: 6 X and u 6 Xo. (ii) lin(dom/*)=X6L. (iii) The function /: X/X0 -»• I, f(x) := f(x), is well defined, fe r(X/X0), /OO(M) = /oo(«) for every u € X, f* = f*\x± and 5£/(£) = 9E/(a;)|Xj. for all x € X and £ 6 R+, where x represents the class of x £ X. Exercise 2.25 Let X be a separated locally convex space and / £ T(X). Assume that there exists u £ X such that / Exercise 2.26 Let X, Y be separated locally convex spaces and F 6 A(X x Y). Assume that F satisfies one of the conditions (ii)-(viii) of Theorem 2.7.1. Prove that the marginal function g : X* —¥ R, g(x") := infy*ey* F*(x*,y*) is convex, iu*-lsc, the infimum is attained for every x* £ X* and goo(u") = mmv*ey(F*)oo(u* ,v*). Moreover, {v* £ Y* \ (F*)oo(0, «*) < 0} is a linear subspace. Exercise 2.27 (Toland-Singer duality formula) Let X be a separated locally convex space, / : X —>• R be a proper function and g £ T(X). Then inf (/(*) - ff(i)) = .inf. (g\x') - f*{x')) • Exercise 2.28 Let X be a separated locally convex space, / £ T(X) be such that 0 € dom/ andp £ P. Consider the following assertions: (i) the mapping P3ti-> t~pf(tx) is nondecreasing for every x £ dom/; (ii) f'(x, —x) +pf(x) < 0 for every x £ dom/; (iii) pf(x) < f'(x,x) for every x € dom/; (iv) {x,x*) >pf(x) for all (x,x*) € grd/; (v) for every x £ int(dom /) there exists x* 6 df(x) such that (x, s*) > pf(x). Prove that (i) <=> (ii) <=> (iii) =$- (iv). Moreover, if / is continuous at 0 then all five assertions above are equivalent. Exercises 151 Exercise 2.29 Let / : X —> IR be a function such that co/ is proper. As sume that for some x € domco/ we have that x = ^2i=i ^»^« an(^ co/(ic) = ^*=1 ~\if{x~i) with Xi G dom/, Ai > 0 for i € ITfc and £)*=1 ^> = 1- Prove that 8c5/(3f) = nti df(xi). Exercise 2.30 Let X be a separated locally convex space and / € A(X). Assume that 0 G dom/ and /|x0 is continuous at 0, where Xo •— aff(dom/). Prove that f = f'(0,x) if xGXo, sup{(x,x*) |x* G<9/(0)W < f'(Q,x) = oo if xGXo\_Xo, { = /'(0,x) = oo if xeX\X0, the supremum being attained for x G Xo. Moreover, Xo is closed if and only if VxGX : /'(0,x)=sup{(x,x*) | x* 6 3/(0)} . Exercise 2.31 Let X be a separated locally convex space and / G A(X) be continuous on int(dom/), supposed to be nonempty. Prove that for all x,y € int(dom/) there exist z £]x,y[and z* G df{z) such that f(y)—f{x) = (y — x, z*). Exercise 2.32 Let X be a separated locally convex space and / G T(X). (i) Consider the conditions: a) df is single valued on dom.3/, b) (df)~1(xl)!~) (df)~1(x2) = 0 for all distinct elements Xi,X2 G X*, c) /* is strictly convex on every segment [xi,X2] C Imdf, d) dom df = int(dom/). Prove that a) <=> b) •<=>• c); moreover, if int(dom/) ^ 0 and / is continuous on int(dom/) then a) => d). (ii) If / is continuous on int(dom/) and (9/)_1 is single-valued on Imdf, show that / is strictly convex on int(dom/). Exercise 2.33 Let X, Y be separated locally convex spaces and / G A(X x Y). Prove that if / is continuous at (xo,j/o) £ dom/, then Prx* (df{x0, j/o)) = df(-,yo)(x0) and Pry (d/(x0, y0)) = d/(x0, -){yo)- Exercise 2.34 Let X be a separated locally convex space and /o,/i 6 A(X). Consider v := inf{/0(x) | /i(x) < 0}, v* := sup inf (/o(x) + A/i(x)), x>o xex with the convention 0 • oo = oo. Suppose that »£R. Prove that v* =v&[rfe>Q : inf{/i(x) | /0(x) < y - e} > 0]. Exercise 2.35 Let {X, \\-\\) be a normed linear space and / : X —> R be a proper convex function. Prove that there exists x* G X* with x* < / if and only if there exists M > 0 such that f(x) + M \\x\\ > 0 for all x G X. 152 Convex analysis in locally convex spaces Exercise 2.36 Let (X, ||-||) be a normed linear space and /i,/2 : X -> R be proper convex functions. Prove that there exist x* 6 X* and a € R such that —/i < x" + a < J2 if and only if there exists M > 0 such that fi(xi) + f2(%2) + M\\xi - x21| > 0 for all on, x2 € X. Exercise 2.37 Let X be a linear space, (Y, ||-||) be a normed linear space, T : X -* Y be a linear operator, yo 6 Y and / : I -> R be a proper convex function. Prove that f(x) + \\Tx + yo\\2 > 0 for all x € X if and only if there 2 exists j/* € y* such that fix) -2(Tx + y0,y*) - ||y*|| > 0 for all x e X. Exercise 2.38 Let (X, ||||) be a normed vector space, C,D C X be closed convex cones and x G X, x* € X*. Prove that + d(x,C) :=inf{||x-x|| | x € C} =max{-(x,x*) | x* 6 Ux* n C } , + + d(x*,C ) =min{||x*-x*|| | a;* 6C } = sup{-(x,x*) | x G C/x n C} , + a,ndsupx€Uxncd(x,D) = supx,eUx,nD+ d(x* ,C ). Exercise 2.39 Let / € A(X) and x € dom/ be such that f(x) > inf/. Consider the sets d Prove that dKf{x) = d-f(x) = [1, oo[-d/(x). The set a Exercise 2.40 Let X be a normed space, (an)n>i C X and (An)„>i C [0,oo[ be such that ^^LjAn = 1. Consider the function /:X->R, f{x) = Y°° A„||x-a„||2. *•—'n=l 1) Prove that the following statements are equivalent: (a) S^Li-^i|la"ll2 < oo (i.e. 0 € dom/), (b) dom/ ^ 0, (c) dom/ = X. 2) Prove that / is finite, convex and continuous when 5Z^Li^illa"l|2 < °°- 3) Suppose that ]C!!!Li^n|lan||2 < °°- Prove that for every x £ X one has 00 m 8f{x) = {2V An||x - On|| x n I x*n e a|| • IKi - a„) Vn € N) . Exercise 2.41 Let X be a normed space and / 6 T(X). Prove that the following statements are equivalent: (a) lim||J.||_too f{x) = 00; (b) [/ < A] is a bounded set for every A > inf^ex /(x); (c) there exists Ao > inf^gx /(x) such that [/ < Ao] is bounded; Exercises 153 (d) there exists a, 0 £ R, a > 0, such that f(x) > a||x|| + /? for every x G X; (e) liminf|)x||_foo /(x)/||x|| > 0; (f) 0£int(dom/*). Moreover, if dim X < oo then the conditions above are equivalent to (c') [/ < iaf /] is nonempty and bounded. Furthermore, if p, q £ ]l, oo[ are such that 1/p + 1/q = 1, the following state ments are equivalent: p (g) liminf|NHoo/(x)/||x|| >0; q (h) limsup||:c,|Hoo r(x")/\\x*\\ < oo; (i) there exists a, /3 £ R, a > 0, such that f(x) > a||x||p + P for every i6X; (j) there exists a,/3 £ R, a > 0, such that /*(x*) < cx||a;*||9 + /? for every a;* eX*. m Exercise 2.42 Let f,fn : R ->• R (n £ N) be convex functions such that (fn{x)) -¥ f{x) for every x £ Rm. Assume that / is coercive. Prove that there m exist a,P £ R with a > 0, no £ N such that /n(x) > a \\x\\ + /3 for all x G R and n > no. Exercise 2.43 Let (X, ||||) be a n.v.s. and C C X be a nonempty closed convex set. Consider the following assertions: (i) there exists XQ G X* such that (x,x*,) > \\x\\ for every x G C; (ii) there exist xo £ -X", X*, £ X* such that (x — xo, x*>) > ||x — xo|| for every x eC; (iii) int(dom sc) 7^ 0; (iv) a) there exists x*, G X* such that {u, x*>) > 0 for every u £ Coo \ {0} and _1 b) for every sequence (xn) C C with (||xn||) —• oo and ( ||x„|| xn) -^ u one has Prove that (i) => (ii) =S> (iii) => (iv). If 0 £ C then (ii) =>• (i). Moreover, if X is a reflexive Banach space then (iv) =>• (ii). Exercise 2.44 Let X be a normed space and / £ A(X) be lower bounded. Consider A > infl€x /(x) =: inf / and p > 0. We envisage the conditions: (a) [f (b) Vx£X : f(x) > inf / + A ~lnf f • max{0, l|a?l| - p}; 2p A f/ (c) Vx'G "2'° t/x» : /*(*') Prove that (a) =>• (b) & (c). Exercise 2.45 Let X be a normed space and / G T(X). Prove the following equivalences: 154 Convex analysis in locally convex spaces (a) liminf||.1.||_+00 /(x)/||x|| >0-»0£ int(dom/*); in this case f(x) liminf 4f-if = sup{/i > 0 | /* is upper bounded on (J.U*}. ||x||-KX> ||X|| (b) liminf ||B|Hoo f(x)/\\x\\ =0«06 Bd(dom/*). (c) liminfHJII-KX, f(x)/\\x\\ < 0 O 0 ^ cl(dom/*); in this case f(x) liminf ~-r{- = -ddom/. (0). llxH-KX) ||Z|| Exercise 2.46 Let p 6 ]1, oo[ and p n f:£ ^R, /((*„)„>i) := Y°° n\xn\ . Prove that / is finite, convex, continuous and lim||„||_+0O f*(y)/\\y\\ — 1. Exercise 2.47 Let X, Y be Hilbert spaces, C C X be a nonempty closed convex set and A G L(X,Y) be a surjective operator. Consider the problem (P) min i||Ac||2, xeC. A solution x of problem (P) is called a spline function in C associated to A. (a) Prove that (P) has optimal solutions if C + ker ^4 is closed. (b) Prove that x G C is an optimal solution of (P) if and only if y := A* (Ax) satisfies the relation (x\y) = min{(x \y) \ x € C}. Exercise 2.48 Let X,Y be Hilbert spaces, C := {x 6 X | Vi, 1 < i < k : (x\a,i) < fii}, where ai,...,ajb G -X', Pi,...,Pk G R and A G £(X,F) be a surjective operator. Consider the problem (P) min 5||AE||2, x G C. Prove that (P) has optimal solutions. Suppose, moreover, that there exists x € X such that (x \ a{) < /3i for every i, 1 < i < k. Prove that lis a solution of (P) if and only if there exists (\i)i A*(Ax) = Aid H V Xkdk andVi, 1 < i < A: : A;((x| a») — /?;) = 0. Exercise 2.49 Let X be a Hilbert space and 01,02,03 be three non colinear elements of X (i.e. they are not situated on the same straight-line). Prove that there exists a unique element x G X such that Vi£l : ||x-Oi|| + ||x-a2|| + ||i-a3|| < ||x - ai|| + ||x - o2|| + ||x - a3||. Moreover, prove that x G co{oi,a2,03}. The point x is called the Toricelli point of the triangle of vertices a\, 02,03. Bibliographical notes 155 Exercise 2.50 Determine the optimal solutions (when they exist) and the value of the problem (Pf) min £ (tx{t) + fiy/1 + (u(t))2) dt, x e Xi, for every fj, € R+ and every i g {1, 2, 3,4}, where Xi:=C[0,l], Xa:= 2/(0,1), 1 X3 := |xeC[0,l] |/o x(t)dt = 0J, X» := {xG 1/(0,1) | /^ x(t)dt = o}. Exercise 2.51 Consider the (convex) optimization problem (P) max J,,1 x(t) dt, xeX, x(0) = x(l) = 0, J,,1 y/l +{x'(t))2dt < L, where L > 0 and X = C^O, 1] := {x : [0,1] —• R | x derivable, x continuous on [0,1]} or X = AC[0,1] := {x : [0,1] -¥ R | x absolutely continuous}. Determine the optimal solutions of problem (P), when they exist, and its value (using, eventually, the dual problem). 2.12 Bibliographical Notes Many results of this chapter are well-known and can be found in several books treating convex analysis: [Rockafellar (1970); Hiriart-Urruty and Lemarechal (1993)] (for finite dimensional spaces), [Ekeland and Temam (1974); Ioffe and Tikhomirov (1974); Barbu and Precupanu (1986); Castaing and Valadier (1977); Phelps (1989); Aze (1997)]. In the sequel we point only those results that are not contained in these books but the last one. Sections 2.1-2.5: The assertion (vii) of Theorem 2.1.5 was established in [Zalinescu (1983b)]. The characterization of convex functions using upper Dini directional derivatives in Theorem 2.1.16 as well as Lemma 2.1.15 are established by Luc and Swaminathan (1993). Theorem 2.2.2 was established by Crouzeix (1980) in finite dimensional spaces and Theorem 2.2.11 by Zalinescu (1978). The notion of quasi-continuous function was introduced by Joly and Laurent (1971); Proposition 2.2.15 and Corollary 2.2.16 are established by Moussaoui and Voile (1997). The notions of cs-convex, cs-closed and cs-complete functions are intro duced by Simons (1990), while that of lcs-closed function by Amara and Ciligot- Travain (1999); all the results concerning lcs-closed functions are due to Amara and Ciligot-Travain (1999). Theorem 2.2.22 (for I = N) and Proposition 2.2.24 can be found in [Kosmol and Muller-Wichards (2001)]; when dimX < oo Theo rem 2.2.22, Corollary 2.2.23 and Proposition 2.2.24 are stated in [Marti (1977)] under the weaker hypothesis that the pointwise boundedness or pointwise con vergence holds on a dense subset of C. Proposition 2.2.17 is stated in [Zalinescu (1992b)]. Proposition 2.4.3 is established by Rockafellar (1966), Theorem 2.4.11 is proved in the book [Rockafellar (1970)] in Rn and stated by Hiriart-Urruty 156 Convex analysis in locally convex spaces (1982) in the general case; one can find another proof in [Aze (1997)]. The local boundedness of df in Theorem 2.4.13 and Corollary 2.4.10 can be found in many books on convex analysis. The assertions (iv) and (vii) of Theorem 2.4.14 are established in [Zalinescu (1980)]; for (i) see also [Anger and Lembcke (1974)]. Theorem 2.4.18 is from the book [Ioffe and Tikhomirov (1974)]. Theorem 2.5.2 was proved by Polyak (1966) under the more stringent conditions that / is a quasi-convex function which is bounded and Lipschitz on bounded sets and at tains its infimum on every closed convex subset of X. Theorem 2.5.5 and Lemma 2.5.3 are established by Borwein and Kortezov (2001), while Lemma 2.5.4 and other results on non-attaining convex functionals are established by Adly et al. (2001a). Sections 2.6-2.9: The systematic use of perturbation functions for cal culating conjugates and subdifferentials was done for the first time by Rock- afellar (1974). The author of this book continued this approach in [Zalinescu (1983a); Zalinescu (1987); Zalinescu (1989); Zalinescu (1992a); Zalinescu (1992b); Zalinescu (1999)]; this permitted, for example, to give simpler proofs to several results stated in [Kutateladze (1977); Kutateladze (1979a); Kutateladze (1979b)]. Theorem 2.6.2(i) was established by Moussaoui and Seeger (1994), but Theorem 2.6.2(h) and Theorem 2.6.3 are new. Corollaries 2.6.4-2.6.7 can be found in [Hiriart-Urruty and Phelps (1993)] and [Moussaoui and Seeger (1994)]; for other results in this direction see the survey paper [Hiriart-Urruty et al. (1995)]. The sufficient conditions for the fundamental duality formula and for the va lidity of the formulas for conjugates and e-subdifferentials are, mainly, those from author's survey paper [Zalinescu (1999)], where one can find detailed historical notes; here we mention only the first use (to our knowledge) of them; actually, all these sufficient conditions are mentioned in that paper, excepting those which use li-convex or lcs-closed functions. So, conditions (iii) and (viii) of Theorem 2.7.1 and the corresponding ones in Theorems 2.8.1, 2.8.3, 2.8.7 and 2.8.10 are the classical ones and can be found in all the mentioned books which treat them. Condition (i) of Theorem 2.7.1 was used by Rockafellar (1974) when Yo = Y and by Zalinescu (1998) in the present form (see also [Combari et al. (1999)]), (ii) and (vi) were introduced in [Zalinescu (1983a)] for Yo = Y (and R replaced by a separated lcs ordered by a normal cone) and in [Zalinescu (1999)] in the present form, (iv) was introduced in [Zalinescu (1992b)] with (Hwi) replaced by (Hx), (v) is stated by Amara and Ciligot-Travain (1999) for X, Y Frechet spaces, $ a lcs-closed function and ,b replaced by lc, (vii) was introduced by Rockafellar (1974) for X, Y Banach spaces (X even reflexive) and Yo = Y, and by Zalinescu (1987) in the present form, while (ix) was used by Joly and Laurent (1971) for $ lsc and by Moussaoui and Voile (1997) for arbitrary 3>; note that condition (b) of [Cominetti (1994), Th. 2.2] implies condition (2.54). Theorem 2.7.4(iv) was established in [Zalinescu (1992b)]; the other conditions were introduced in [Zalinescu (1999)] (but in (v) it was assumed that F is lsc and C is closed). Condition (iii) of Theorem 2.8.1 was introduced in [Ekeland and Temam (1974)], (vii) was introduced in [Zalinescu (1984)] (for Y0 = Y), (iv) was intro- Bibliographical notes 157 duced in [Zalinescu (1992b)], (i) and (viii) in [Zalinescu (1998)] for X,Y normed spaces, while conditions (ii) and (vi) were introduced in [Zalinescu (1999)]. Condition (vii) of Theorem 2.8.3 was introduced by Borwein (1983) for Yo = Y (see also [Zalinescu (1986)]), (x) was introduced by Borwein and Lewis (1992), (ix) was introduced by Moussaoui and Voile (1997), (i) by Zalinescu (1998) for X, Y normed spaces, (ii) was introduced by Combari et al. (1999), (iv) was introduced by Zalinescu (1999) (a slightly stronger form was used in [Simons (1990)]) as well as conditions (v) and (vi). For C G L(X, Y), generally, Theorem 2.8.6 is obtained under the correspond ing conditions of Theorem 2.8.3 (taking / = 0). For C a densely defined closed linear operator, Rockafellar (1974) and Hiriart-Urruty Hiriart-Urruty (1982) ob tained the results for g lsc, X, Y Banach spaces and Yo = Y (in [Rockafellar (1974)] X is reflexive and e = 0); Aze (1994) obtained the formula for the conju gate under condition (ii) in normed spaces. For general C condition (iv) was used by Zalinescu (1992b); the other conditions (excepting (v)) were used in [Zalinescu (1999)]. Condition (ix) of Theorem 2.8.7 was used by Joly and Laurent (1971), (vii) for X a Banach space was introduced by Attouch and Brezis (1986), (vi) was in troduced by Zalinescu (1992b), (i) was introduced by Aze (1994) in an equivalent form in normed spaces, while (ii) was introduced by Combari et al. (1999). Corollary 2.8.8 was obtained by Voile (1994) for X a Banach space, /, g lsc and '* replaced by !C. Formula (2.65) from Proposition 2.8.9 was obtained by Aze (1994) and Simons (1998b) for x = 0 € (dom/ — domg)1 when /,g are lsc. The case / = 0 of Theorem 2.8.10 was considered by several authors; condi tion (iii) was used in [Hiriart-Urruty (1982); Lemaire (1985); Zalinescu (1983a); Zalinescu (1984)]; the algebraic case was considered in [Kutateladze (1979a); Kutateladze (1979b); Zalinescu (1983a)]; Zalinescu (1984) considered a stronger version of (v) (for lsc functions and Yo = Y); note that in these papers g is as sumed to be Q-increasing on the entire space. The general case was considered in [Combari et al. (1994)] under (iii) and a condition stronger than (v) (/, g, H lsc and lc instead of lb) and in [Combari et al. (1999)] under condition (i) and a variant of (iv); it was also considered by Moussaoui and Voile (1997) under condition (vii). The formula for d(p(x) in Corollary 2.8.11 can be found in [Combari et al. (1994)]. The formula for d Section 2.10: The minimax theorem is also classical and can be found in the books [Barbu and Precupanu (1986); Simons (1998a)]. Exercises: Exercise 2.2 is from [Penot and Bougeard (1988)], Exercise 2.3 is from [Crouzeix (1981)], Exercise 2.6 is from [Marti (1977)], Exercise 2.15 is from [Crouzeix et al. (1992)] (there in a more complete form), Exercises 2.18 and 2.29 are from [Hiriart-Urruty and Lemarechal (1993)], Exercise 2.19 is from [Lions and Rochet (1986)], Exercise 2.20 is the celebrated cancellation lemma from [Hormander (1955)], Exercise 2.21 is from [Hiriart-Urruty (1998)], Exercise 2.22 is from [Jourani (2000)], the assertions in Exercise 2.23 are well-known (see also [Aze (1997)]), Exercise 2.25 can be found in [Borwein and Kortezov (2001)] (in normed spaces), the formula for goo under condition (vii) of Theorem 2.8.1 in Exercise 2.26 is obtained in [Amara and Ciligot-Travain (1999)] and [Amara (1998)], Exercise 2.27 is the well-known Toland-Singer duality formula (see [Toland (1978)] and [Singer (1979)]), Exercise 2.30 is from [Zalinescu (1999)], the equivalence of a) and c) (for Banach spaces) in Exercise 2.32 can be found in [Bauschke et al. (2001)] (see also [Barbu and Da Prato (1985)]), Exercises 2.33 and 2.34 are from [Eremin and Astafiev (1976)], Exercises 2.35, 2.36 and 2.37 are from [Simons (1998a)], the last formula in Exercise 2.38 is from [Walkup and Wets (1967)], Exercise 2.39 is from [Penot (1998a)], the formula for the subdifferential of the function considered in Exercise 2.40 and its proof are from [Aussel et al. (1995)], the equivalences (e) •£> (f) and (g) & (h) of Exercise 2.41 are from [Zalinescu (1983b)], the equivalence of conditions (ii)-(iv) in Exercise 2.43 are established in [Adly et al. (2001b)] in reflexive Banach spaces, a weaker variant of the implication (a) => (c) of Exercise 2.44 is stated in [Aze and Rahmouni (1996)], Exercise 2.45(c) is from [Borwein and Vanderwerff (1995)], Exercises 2.47, 2.48 axe from [Laurent (1972)]. Chapter 3 Some Results and Applications of Convex Analysis in Normed Spaces 3.1 Further Fundamental Results in Convex Analysis Throughout this chapter X, Y are normed spaces and X*,Y* are their duals endowed with the dual norms. As an application of Ekeland's variational principle and of some results relative to convex functions, we state the following multi-propose general ization of the Br0ndsted-Rockafellar theorem. Theorem 3.1.1 (Borwein) Let X be a Banach space and f £ r(X). Consider e £ P, XQ £ dom/, XQ £ def{xa) and /3 € K+. Then there exist xe £ X, y* € Ux- and Ae £ [—1, +1] such that \\xe - Xo\\ + P • \(X£ - X0,X*0)\ < y/E, (3.1) x% := x*0 + VE(y* + /3\ex*) £ df{xe). (3.2) Moreover ||*e - soil < Ve, \te-xl\\ \(xe-x0,x*)\ x*e£d2ef(x0), \f(xe)-f(x0)\ || • Ho •. x-• K, \\x\\0~\\x\\ + p-\(x,x*)\, is, obviously, a norm on X, equivalent to the initial norm. Therefore {X, ||-||0) is a Banach space. Consider the function g := / — XQ. It is obvious 159 160 Some results and applications of convex analysis in normed spaces that Xo £ dom WxGX : g{x)>g(x0)-e [& x*0 £ dsf{x0)]. We apply Ekeland's theorem (Theorem 1.4.1) to g, y/e and the metric d given by d(x,y) := \\x — y\\o. So there exists xe £ domg such that g{xE) + y/e- \\x£ -x0\\o < g(x0), (3.6) \/x£X, x^xe : g(x£) < g(x) + y/e • \\x - xe\\0. (3.7) From Eq. (3.6) we obtain that g(xe) + Ve(\\xs - ar0|| +0-\{xe- x0, XQ)\) < g(x0) < g(xe) + e, whence Eq. (3.1) follows immediately. Let us consider the function h : X -> E, h(x) := g(x)+y/e-\\x-xE\\0 - f{x)-{x,xl)+y/E-\\x-xE\\+Py/e-\(x-xe,xl)\; by Eq. (3.7) xe is a minimum point of h. Therefore 0 £ dh(xe). Since h is the sum of four convex functions, three of them being continuous, and taking into account the expression of the subdifferential of a norm (Corollary 2.4.16) and of the absolute value of a linear functional (at the end of Section 2.8), we have that 0 £ dh(xe) = df(x£) - x* + y/i • Ux- + PVe • [-1, +1] • x%. Therefore there exists x* £ df(xs), y* £ Ux* and Xe £ [—1,1] such that Eq. (3.2) is verified. The estimations from Eq. (3.3) follow immediately from Eqs. (3.1) and (3.2). Using again Eqs. (3.1) and (3.2) we obtain that |(a;e -x0,x* -XQ)\ = y/e-\{xE - xQ,y* + p\Ex%}\ < v/i(||ze - x0\\ • W\ + 0\Xe\ -\(xe- x0,x*0)\) < y/e(\\xs - x0\\ + fi • \{x£ -x0,x*0)\) From Eqs. (3.1) and (3.8) we get |(a;e -x0,x*)\ < \{xe -x0,x* -x^)\ + \(x£ -X0,XQ)\ i.e. the estimation in Eq. (3.4) holds, too. Since x$ G dEf(xo) and x* G df(xE), we get (X0 ~Xe,X*£ -XQ) + (X0 -X£,X*Q) = (X0 ~Xe,X*) < f(x0) ~f(Xe) < (X0 -XE,XQ) +£. Using relations (3.1) and (3.8), we obtain that \f(x0) - f(xe)\ < \(x0-xe,x*0)\+s XQ G dEf(x0), x* G df(xE), we obtain that (x-X0,X*e) = (X-X£,X*) + (Xe -X0,X* -XQ) + (XE -X0,XQ) < f{x) - f(xe) +s + f(x£) - f{x0) + e - f(x) - f(x0) + 2e for every x G X, i.e. x£ G d2ef(x0); hence the first relation in Eq. (3.5) holds, too. • The next result is well-known. The first part is an immediate conse quence of Borwein's theorem, while the density part, which follows easily from the first one, will be reinforced in Theorem 3.1.4 below. Theorem 3.1.2 (Br0ndsted-Rockafellar) Let X be a Banach space and f € r(^)- Consider e > 0 and (XO,XQ) G grd6f. Then there exists (x£,x*) G gr<9/ such that \\x£— XQ\\ < ^Jl and \\X*—XQ\\ < y/e. Inparticular domf C cl(dom<9/) and dom/* C cl(Im<9/). D Another consequence of Borwein's theorem is the following result which will be completed in Proposition 3.1.10 below. Proposition 3.1.3 Let X be a Banach space and f G T(X). Then f(x) = sup^z - z,z*) + f(z) I (z,z*) G grdf} = sup{(x, z*) - /•(**) | z" G lm(df)} (3.9) for every x G dom/. Proof. The second equality in Eq. (3.9) is obvious because f(z)+f*(z*) = {z, z*) for any (z, z*) G grdf. Let x G dom / be fixed. When (z, z*) G grdf we have (a; — z, z*) + f(z) < f(x) for every i£l; hence f(x) > sup{(a; - z,z*) + f(z) | [z,z*) egrdf}. Because / is lsc, by Theorem 2.4.4 (hi), there 162 Some results and applications of convex analysis in normed spaces exists x* € de/2f(x). Applying Borwein's theorem we get {xe,x*) £ grdf such that x* 6 def(x). Therefore (xe — x,x*) < f{xe) - f(x) + e, whence f(x) — e < {x — xe,x*) + f(xe). Hence relation (3.9) holds. • As announced before, the density results in Theorem 3.1.2 can be stated in a stronger form. In the next result (xn) —>/ x means that (a;n) —> x and x {f( n)) -* f(x), and similarly for (xn) ->•/• x*. Theorem 3.1.4 Let X be a Banach space and f € r(X). Then (i) Vx edom/, 3((xn,xn)) Cgrdf : (a;„) -+/ x; (ii) Vx* E dom/*, 3 ((x„,<)) C gvdf : «) -•,. x*. Proof, (i) Consider x € dom/. Because / is lsc, by Theorem 2.4.4 (hi), for every n S N there exists x* € dn-2f(x). Applying Borwein's 2 theorem for (x,x*), e = n~ and /? = 1, we get (xn,xn) € grdf such that x _1 2 x \\ n — x\\ < n , \f(xn) — f(x)\ < n~ + n~ . Hence (xn) -^/ x. (ii) Because /* is lsc, it is sufficient to show that for every e > 0 there exists (y, y*) e df such that \\y* - x*\\ 0. Consider also the function g := fO(x* + irux) € A(X). Then g(x) > (fnx*)(x) = (x,x*) - f*(x*) for every x € X. It follows that g € T(X) and f(x) + (—x,x*) > g~(0) > —f*(x*). By Proposition 3.1.3 there exists (z,z*) € dg such that -g*(z*) = g(z) - (z,z*) > -f*(x*) - e/2. But g*(z*) = f*(z*) + (x* + trUx)*(z*) = f*(z*) + r l|ar* - z*\\. Therefore f*(z*) + r\\x*-z*\\ limsup(/(zn) + (u„,a;*» Setting en := f(z„) + f*{z*) — (zn,z*), we have that 0 < £„ = f{Zn) + (U„, X*) - (Zn + Un, Z*) + (Un, Z* - X*) + f*(z*). Taking into account that (un,z* — x*) < r \\x* — z*\\, from Eq. (3.10) we obtain that limsup£n < 0. Hence lime„ = 0. Because (un) is bounded, so Further fundamental results in convex analysis 163 is (zn). Therefore there exists r' > 0 such that (zn) c r'Ux- Take nEN so that S := en < 1 and VS(r' + 2)(1 + \\z*\\) < e/2; set z := zn. Of course, we have that z* G dsf(z). By Borwein's theorem (for /? = 1) there exists (y,y*) G df such that \\z - y\\ < y/S, \\z* - y*\\ < VS{1 + ||z*||) and |/(«) ~ f(v)\ /*(»*) = = (y,y* - z*) + (y-z,z*)-6 + f*(z*) + f{z) - f(y) < VS(V5 + r') (1 + ||z*||) + \f5 \\z*\\ -5 + f(x*) +s/2 + 6 + V6 < f*(x*) + e/2 + VS(r' + 2)(1 + ||«*||) < f*(x*) + e. The proof is complete. • Using Br0ndsted-Rockafellar's theorem we can add other sufficient con ditions for the validity of the conclusions of Theorems 2.8.1 and 2.8.7. Proposition 3.1.5 Let X,Y be Banach spaces, F £ T(X x Y), A G L(X,Y), D = {Ax - y \ (x,y) € domF} and E = {Ax - y \ (x,y) € dom&F}. Then icE = iiE = ic(coE) = ri(coE) = icD = riD. Moreover, if one of the above sets is nonempty then icE = E — D; in particular the sets ICE and E are convex. Proof. Consider the linear operator T : X xY ->• Y defined by T(x, y) := Ax — y. Since domdF C dom.F and domF is convex, we have that E = T(domdF) CcoE = T(co{domdF)) CD = T(domF). By Theorem 3.1.2 we have that domF C domdF. Hence, using the conti nuity of T, we have that E C D C T(domdF) C T(domdF) = E. Using the properties of the affine hull (see p. 2), it follows that aff E C aff D c aff E. Therefore, if aff E is closed then aff D — aff E; the above relation shows that %CE c %CD. Let us show the converse inclusion. Let %C 2/o G D and consider the function F0 6 T(X x Y), F0(x,y) :— F(x,y — yo)- Of course, domF0 = domF + (0,j/o) and dom9F0 = dom9F + {0,y0). ic Therefore 0 € (Pry(domF0)). Taking such that dF0(x,Ax) = dF(x,Ax — y0) ^ 0. Therefore yo € E. So we obtained that *D = ICD C E C D, which shows that aff E = aff-D and *D C iE. It follows that icE = icD = ic(coE). As observed after the proof of Theorem 2.8.1, in our situation, if %CD is nonempty we have that icD = rintD. Suppose that %CD 9^ 0 (for example). From what was proved above, we have that rinti? = lD = icD = rintE = lE = icE C E C D C E. Hence E~ = T°E = D:. The conclusion follows. • Remark 3.1.1 Taking into account the preceding proposition, for X, Y Banach spaces and F G T(X x7), we may add to the sufficient conditions in Theorem 2.8.1 the following conditions: 0 G ri{Ax - y \ (x,y) £ domdF}, 0 £ ic{Ax -y\(x,y)€ domdF}, 0 € ic{Ax -y\(x,y)e co(domdF)}, y0 = cone{ Ar - y | (x, y) 6 dom OF} is a closed linear space. Indeed, the first three conditions are, evidently (using the preceding proposition), equivalent to 0 £ lc{Ax — y \ (x,y) £ domF}. In the fourth lc case y0 = aff {Ax — y\(x,y)e co(dom OF)}, and so 0 e {Ax — y\(x,y)e domF}. An important particular case of the preceding proposition is when X = Y, A = Idx and F(x,y) = f(x) + g(y) with f,g 6 T(X); in this case D — dom/ — domg and E = domdf — domdg. Using the Borwein theorem one obtains a formula for the subdifferential of a composition g o A of a lsc convex function and a continuous linear operator without qualification conditions. Theorem 3.1.6 Let X,Y be Banach spaces, A e C{X,Y), g £ T(Y), f = g o A and x 6 dom/. Then x* 6 df(x) if and only if there exists a net ((yi,yt))ieI C grds such that (yt) ->• y := Ax, (g(yi)) ->• g{y), ((yi-y,y*))^Oand(A*y*)^x*. If X is reflexive one can take sequences instead of nets and impose norm convergence instead of w* -convergence. Further fundamental results in convex analysis 165 Proof. Let (fa,y*))i€l C gidg be such that fa) -> y, (fa - y,y*)) ->• 0, {gfa)) -> g(y) and (i4'i/r) ^ x*. Then (y - yhy*) < gfa-gfa) for all i G I and t/SF. It follows that (x' - x,A*y*) + (y- yhy*) = (Ax' - yuy*) < f(x') - gfa) for alH G / and x' G X. Taking the limit we obtain that (x' — x,x*) < /(x') - f(x) for all x' G X, and so x* G d/(x). Let now x* G df(x) and consider (£„)„6N 4- 0- Using Corollary 2.6.5 we have that x* G w*-clA*(d£ng(y)) for every n G N. Let A/" be a base of w*- neighborhoods of x* and consider / = Nx M with (n, V) y (n1, V) iff n > n' and V CV. Then for all i = (n, V) G / there exists z* G dEng(y) such that A*£* G V. Taking /? = 1 and e = e^ in Theorem 3.1.1, there exists (yi,y*) G 3g such that \\yi - T/|| < e„, ||y* - ^|| < en, \g(yt) - g(y)\ < en(en + l) and \(Vi ~ V,V*)\ < en(en + 1). Because \\A*z* - A*y*\\ < \\A*\\ • \\y* - z*\\ < en \\A*\\ and (A*z*)ieI ^> x*, we obtain that (A*y*)i€l ^> x*. If X is reflexive then w*-cli*(9fBg(j/)) = cl A* (dSng(y)), and so, for every n G N, we can take z*n G dEng(y) such that ||A*z* — x*|| < e„. The conclusion follows. • Using the preceding result one obtains a similar formula for the sub- differential of the sum of two lsc proper convex functions. Theorem 3.1.7 Let X be a Banach space, /i,/2 G T(X) and x G dom /i fl dom /2. Then x* eSf/i+M (x) if and only if there exist two nets x x {( k,i> t,i))ieI C gr<9/fc, k = l,2, such that (xM)ie/ ->• x, (fk(xk,i))ieI -> /fc(x), ((xM -x,x^i)).e/ -^0 for k = 1,2 and (x*^ + x*2 3.1.6 there exists a net ((yi,y*))ieI C gvdg verifying the conditions of the theorem with y := (x,x). Taking yt = (xi)j,x2,i) and y* = (x'^x^j) and taking into account that dg(x\,X2) — dfi(x{) x 0f2(x2), we have that ((xk,i,x*ki))ieI C gvdfk and (xk,i)iei -»• x for k = 1,2. Assume that x (fi(xi,i)) A fi( )- Because /1 is lsc at x, limsupi6/ /i(xi,i) > /i(x). Thus 166 Some results and applications of convex analysis in normed spaces there exists S > 0 such that J := {i £ I \ fi(xiti) — fi(x) > 5} is co- final. It follows that g(yi) - g(y) > 6 + (f2{x2,i) — f2{x)) for all i £ J. Taking the limit inferior, we obtain the contradiction 0 > S. Therefore {fk(%k,i))ieI -^ fk(x) for k — 1,2. The inequalities fi(xi,i) - fi(x) < (xij -x,x{A) < (yi -y,y*) - {h(x2,i) - f2(x)) for i £ I imply that ({xij — x,x{ti}) -> 0. D Using Br0ndsted-Rockafellar theorem we obtain another famous result. Theorem 3.1.8 (Bishop-Phelps) Let X be a Banach space and C C X, C 7^ X, be a nonempty closed convex set. Then (i) The set of support points of C is dense in BdC. (ii) The set of support functionals of C is dense in the set of continuous linear functionals which are bounded above on C. Moreover, ifC is bounded, then the set of support functionals of C is dense in X*. Proof, (i) Let x0 6 Bd C and e £ ]0,1[. Taking into account that C ^ X, 2 there exists xi £ X \C such that \\xi — x0|| < £ - Applying a separation theorem, there exists XQ £ Sx> such that SVLPX£C{X,XQ) < (XI,XQ). SO, for every x £ C, 2 {x - X0,XQ) = (x - XI,XQ) + (xi - X0,XQ) < (x - xi,xl) + \\x\ - x0\\ < e , whence XQ £ de2tc(xo)- Applying the Br0ndsted-Rockafellar theorem we have that 3xe £ C, 3a;* £ dbc{xe) : \\x£ - x0\\ < e, \\x* - XQ\\ < e < 1. Since \\XQ\\ = 1, we have that x*e ^ 0, and so xE is a support point of C with \\x£ — xo\\ < e. The conclusion holds. (ii) Using the second part of Theorem 3.1.2 we have that dom(tc)* C cKJmdic) = cl(Imdtc \ {0}), which shows that the conclusion of the theorem is true. • The following theorem will be used for a simple proof of the Rockafellar theorem. Further fundamental results in convex analysis 167 Theorem 3.1.9 (Simons) Let X be a Banach space, f € T(X) and x € X, n G R be such that inf / < 77 < f(x). Consider L := sup -fizz rr, x€X\{x} ||ar — ar|| and dx • X -V R, dj(x) := ||i — x||. Then: (i) 0 < -L < oo and inf (/ + Ldx) > n; (ii) Vee]0,l[,3i/6X : (/+ L«fc)(i/) < inf(/+ id*) + eL||i- y\\; (iii) Ve€]0,l[, 3 («,*•) Ggrd/ : {x - z,z*) > (I - e)L\\x - z\\ > 0 and\\z*\\ < (l + s)L; (iv) Ve €]0,1[, 3(z,z*) € grS/ : (z - z,z*> + /(*) > 17 and L < \\z*\\<(l + e)L. Proof, (i) Because 77 > inf /, it is clear that L > 0. Since 77 < /(a;) and / is lsc at x, there exists p > 0 such that /(x) > 77 for every a; € B(x,p). Furthermore, since / € r(X), there exist x* € X* and a 6 R such that f(x) > (a;!a;*) ~ Q f°r every a; £ X. Therefore 77 — f(x) V* £ X, \\x - x\\ > p : 5_jlZg) < ||x*|| + _]_ < li^H + 2. IF ~ x\\ \\x ~ x\\ P This relation and the choice of p show that L < 00. From the expression of L we obtain immediately that inf (/ + Ldx) > 77. (ii) Let e e]0,1[. Since (1 - e)L < L, from the definition of L there exists y £ X, y ^ x, such that (77 — f{y))/\\x — y\\ > (1 — e)L, and so (/ + Ldx)(y) < 77 + eLdi(y) < inf(/ + Ldx) + eL\\y - x\\. (iii) Let e €]0,1[ be fixed. The function / + Ldx 1S proper, lsc and bounded from below, while the element y from (ii) is in dom(/ + Ldx) = dom/. Taking the metric don X defined by d(xi,X2) := L\\xi—X2\\, (X,d) is a complete metric space. Using Ekeland's variational principle we get the existence of z € X such that (/ + Ldx)(z) + EL\\Z - 7,11 < (/ + Ldx)(y) 168 Some results and applications of convex analysis in normed spaces and (f + Ldw)(z) < {f + Ld¥)(x) + eL\\x - z\\ VxeX. The first relation and (ii) give ||^ — yj| < \\x — y\\, and so z ^ I. The second relation says that z is a minimum point of the function / + Ld^ + sLdz. Taking into account that cfe- and dz are continuous convex functions, it follows that 0 G d(f + Ldw + eLdz)(z) = df(z) + Ldd^{z) + eLddz(z). But ddz{z) = d\\ • ||(0) = Ux- and dd^(z) = {x* € Ux* | (z - x,x*) = \\z — x\\}. Hence there exist z* G df(z) and x*, y* £ Ux* such that z* = Lx* + sLy* and (x - z, x*) — \\x - z\\. Therefore ||z*|| < (1 + e)L and (x — z, z*) = {x — z, Lx* + eLy*) = L\\x — z\\ + eL(x — z, y*) > L\\x - z\\ - eL\\x - z\\ = L(l - e)\\x - z\\ > 0. (iv) Let e £]0,1[ be fixed and consider e' := e/3. Let us take M := (1 + 2e')L. Since / + Md% > f + Ld% > r], we can apply Ekeland's theorem for / + Md-x, an element xo of dom /, e' > 0 and the metric defined at (iii). We get so the existence of z e X such that Viel: (f + Mtk)(z)<(f + Mds)(x)+e'L\\x-z\\. As in the proof of (iii), there exist z* € df(z), x*, y* € Ux* such that z* = Mx* +e'Ly* and (x-z,x*) = \\x - z\\. Thus ||z*|| < (1 + s)L and (x - z, z*) > (M - e'L)\\x - z\\ = (1 + e'L)\\x - z\\. So, for x = z we have that (x — z, z*) + f(z) = f(x) > rj, while for ~x yi z we have (x-z,z*) + f(z) > (l + e')L\\x-z\\ + f(z) = (f + Ldw)(z)+e'L\\x-z\\ > v. Therefore (x — z, z*) + f(z) > T). Moreover ||i - z|| • \\z*\\ >(x-x,z*) = [(x - z, z*) + f(z)] - [(x - z, z*) + f(z)] >V~ f(x) for every x € X; the last inequality holds because z* G df(z). Dividing by ||af — x\\ for x 7^ x, we obtain that ||z*|| > L. D Convexity and monotonicity of subdifferentials 169 Using the preceding theorem we reinforce slightly Proposition 3.1.3. Proposition 3.1.10 Let X be a Banach space and f 6 T(X). Then relation (3.9) holds for every x 6 X. Proof. Taking into account Proposition 3.1.3 we have to show that oo = sup{(a; — z,z*) + f(z) \ (z,z*) € gr<9/} when x £ dom/. So, let x £ X \ dom/ and take inf / < 77 < 00. Using Theorem 3.1.9 (iv), there exists {z,z*) E gr<9/ such that (a; - z,z*) + f(z) > rj. The conclusion follows. • The maximal monotone operators are of a great importance in the the ory of evolution equations. A significant example of such operators is the subdifferential of a proper lsc convex function on a Banach space. Theorem 3.1.11 (Rockafellar) Let X be a Banach space and f 6 T(X). Then df is a maximal monotone operator. Proof. Let (x,x*) EXxX*\gvdf. Then x* £ df{x), and so 0 i df(x), where / := / — x*. It follows that inf/ < fix). Applying assertion (iii) of Simons' theorem, there exists (z, z*) € gr9/ such that (x — z,z*} > 0. Consider z* := z* + x*\ we have that (z, z*) £ df and (z -x,z*— x*) < 0, which shows that the set df U {(x,x*)} is not monotone. Therefore df is a maximal monotone operator. • 3.2 Convexity and Monotonicity of Subdifferentials The aim of this section is to show that the monotonicity of an abstract sub- differential of a lower semicontinuous function ensures its convexity; among these abstract subdifferentials one can mention the Clarke subdifferential we introduce below. Throughout this section (X, ||.||) is a normed vector space. Consider |/McI and x e clM. In the sequel we shall denote by (xn) ->M X a sequence (xn) C M with (xn) -» x; more generally, x ->M X will mean x £ M and x —> x. The Clarke's tangent cone of M at x is defined by Tc{M,x):={ueX\V{tn) 10, V(xn)-*Mx, 3(un)^u, VneN : xn + tnun 6 M). Recall another cone introduced in Section 2.3: C(M,x) := elf|J t-(M-x)) =cone{M-x). 170 Some results and applications of convex analysis in normed spaces Because clM -x C C(M,x) C C(clM,i), we have C{M,x) = C(clM,:r). Note (Exercise!) that 0ETc(M,x)cC(M,x). (3.11) Several properties of the tangent cone in the sense of Clarke are collected in the following proposition. Proposition 3.2.1 Let 0 ^ M C X and xEc\M. Then: (i) Tc{M,x) is a nonempty closed convex cone; (ii) TC(M, x) = Tc(cl M, x) = Tc(M D V, x) for all V E M{x); (iii) if M is a convex set then Tc(M,x) = C(M,x). Proof, (i) It is obvious that Xu E Tc(M,x) for A > 0 and u E Tc(M,x); hence Tc(M,x) is a cone. Let u,v € Tc(M,x) and consider (tn) 4- 0 and (xn) ~>M x. Then there exists (un) -> u such that x'n := xn + tnun G M for every n E N. Of course (xJJ —>• 3;. Hence there exists (vn) —> u such that £n + in(w„ + u„) = x^ + inwn G M for every n. Therefore u + v E Tc(M,x). k fc Consider now (u )ke^ C Tc(M,x) with (u ) -+ u E X. Let us show fc that u E Tc{M,x). For this take (tn) I 0 and (xn) ->M £• Because u E fc k Tc{M,x), there exists (u*) ->• u such that xn + tnu n E M for every n G N. k Hence xn+tnu n E M for all k, n E N. For every n EN there exists k'n E N k l such that ||u* — u \\ < n~ for every k > k'n. Let (fc„) C N be an increasing k _1 sequence such that kn > k'n; then ||u* - u \\ < n for all k,n E N with n k n k > A;„. Let u„ := u* . Of course, £„ + £nu„ = xn + tnu n G M for every n. n n kn kn _1 n As ||M„ - u|| = \\un - u|| < ||u* - u || + \\u - u\\ < n + ||u* - u||, we have that (un) ->• u. Hence u G Tc(M,x). Therefore Tc(M,x) is a closed convex cone. (ii) Let u E Tc(M,x) and take (tn) I 0 and (xn) ->CIM £• For every n G 1 N there exists xn E M such that \\xn — xn\\ < tn/n, i.e. xn = xn + tnn~ u'n with u'n E Ux- Hence (xn) ->M x. Therefore there exists («„) -> u with l xn + tnun = xn + tn(n~ u'n + un) G M C clM for every n E N. As (n_1u^ + u„) —> u, we have that u G Ib(clM,x). Conversely, let u G Tc(c\M,x) and take (tn) I 0 and (zn) -tM x. There exists (un) -> u with xn + tnun G cl M for every n. Like above, for l every n E N there exists u^ G Ux such that a;„ + tnun + tnn~ u'n G M. Because (u„+n_1u^) —> ii, we have that w E TQ^M^ X). Hence TQ^M, X) — Tc(dM,x). Convexity and monotonicity of subdifferentials 171 The equality Tc{M,x) = 7b(M n V,x) is obvious when V is a neigh borhood of x. (hi) Let M be convex. Taking into consideration (ii) we may assume that M is closed. Consider x E M and take (£„) 4- 0 and (xn) —»M X. There exists no € N such that tn < 1/2 for n > no. Take un := x + x - 2xn for n > n0 and un = 0 otherwise. Of course, (un) -¥ x — x. As xn + tnun = (1 — 2tn)xn + tnx + tnx S M for n > no, we have that x-x € Tc{M,x). It follows that C(M,x) C Tc(M,x). Using Eq. (3.11) we obtain the conclusion. D From (ii) and Eq. (3.11) we obtain that Tc(M,x) C n C(MHV,x) =: TB{M,x); ' 'V'gJV(x) TB{M,X) is the well known tangent cone in the sense of Bouligand of M atxed M. Let now / : X —> M. be a proper function and x 6 dom /. It is natural to consider the tangent cone Tc (epi/, (x, f(x))). This cone is related to the Clarke—Rockafellar directional derivative of / at x introduced as follows: ,*,_ . , . „ fix + tv) — a f'{x,u):=sup limsup mi e>0 tiQ,(x,a)->epi,(x,f(x)) ll«-«ll It follows that , ,_ , . , f{x + tv) - f(x) /'(a;,w)=sut p inf sup mi , E £>0 <5>0 o /t(x,u)=sup limsup inf /(*+ tV} ~~f{x) (3.12) e>0 (4.0, x->/5 ll«-«ll Proposition 3.2.2 Let f : X —> R be a proper function and x G dom/. Then epi ft(x, •) = Tc (epi /, (x, f(x))). In particular ft(x, •) is a Isc con vex function; ft(x, •) is a Isc sublinear function if and only if ft(x, 0) = 0. Moreover, if f is convex then epi ft(x, •) = cl (epi f'(x, •)). Proof. Let (u,A) G Tc (epi f, (x, f(x))); assume that ft(x,u) > A, and take ft(x, u) > A' > A. Then there exist EQ > 0, ((xn,an)) -»epi/ (x,f(x)) and (tn) 4- 0 such that inf„6c(U)£o) t~* (f(xn + tnv) — an) > A'. Because (u, A) 6 Tc (epi f, (x, f(x))), there exists the sequence ((un,Xn)) converg ing to (u, A) such that (xn,an) + tn(un,\n) G epi/, i.e. f(xn + tnun) < &n + tnXn for all n G N. Since (un) -» u, there exists no such that un G D(u,eo) for n > no- Hence v^ -r f(xn + tnv)-an f(xn + tnun)-an A < inf — < — < A„ Vn > n0. ||w—u|| Let now f^(x~,u) < A and take ((xn,an)) —>epif (x,f(x)) and (tn) I 0. Let us fix (ek) I 0. Because • • f(x + tv) - a in fr sup in tf — < A 5>0 0 There exists n'k G N such that 0 < t„ < 8k, \\xn — x\\ < 8k, \an — f(x)\ < 1 8k for all n > n'k. Therefore inf„€£>(U)efc) t" (f(xn + tnv) - an) < A, which shows that for every k and every n>n'k there exists u* 6 D(u, e^) such that f(xn + tnUn) < a„ + Xtn. Consider an increasing sequence (n^) C N such that nk > n'k for every k G N. Taking un := u* for n^ cone (epi/ - (x, f(x))) C epi f'(x, •) C cone(epi/ - (x, f(x))), Convexity and monotonicity of subdifferentials 173 we have that epip(x, •) = C (epi/, (x,f(x))) = cl(epi/'(:r, •))• D + Note that when / is not lsc at x, f^(x, •) and / (x, •) may be different, where, as usual, / is the lower semicontinuous envelope of /. Take for example / : R ->• R, f(x) = 0 for x ^ 0, /(0) = 1; /t(0, •) = -co, but / (0, •) = 0. When / is lsc at x then f*(x, •) and / (x, •) coincide. When / has additional properties, Z1" has a simpler expression. Proposition 3.2.3 Let f : X —> R be a proper function and x 6 dom/. Then for every u € X we have: (i) limsupj.^. xP(x,u) < f^(x,u); moreover, if f is lsc atx then ,-lv- N / • r • f(x + tv)- f(x) f'(x,u)0 S>0 |j^—^||< < inf sup . (3.13) *>° \\z-x\\ (ii) // / is continuous at x then ,-iv- x • r . t f{y + tv) - f(x) f'{x,u)— sup inf sup mi . e>0 <5>0 \\x-x\\ (iv) If f is finite and Gateaux differentiate on B(x,r), and V/ is con tinuous at x then f^(x, •) = V/(3;). Proof. Throughout the proof u € X is a fixed element. (i) Let f*(x, u) < X and consider e > 0; there exists 6 > 0 such that . f(x + tv)-f(x) ^ sup inf < A. \\x-x\\<8,0 Take 6' := 6/2. Let x' e £(x,<5') with \f{x') - f(x)\ < 6' and x, t be such that \\x - x'|| <6',0 0 < t < 8 and f(x) < f(x) + 8. It follows that sup {mt^D^r1 (f{x + tv) - f(x)) | t G ]0, 8'}, x e D{x', 8'), f(x) sup inf ^HM> sup inf ^ + ^~^, 0<£<<5 llu-ull^e t 0 A. Let 8 :=mm{80,r/(l+e + \\u\\)}. Then for all x £ B(x,8), t e]0,8] and v e D(u,e) we have that \\x + tv - x\\ < 8 + S\\v\\ < 8(1 + e + \\u\\) < r. Therefore for such x, t, v we have f(x + tv) - f{x) < f(x + tv) - f(x + tu) f{x + tu)-f{x) Hence . , fix + tv) - fix) fix + tu) - fix) sup inf ^ '-—i±-± < sup — —i-^-L + Le, \\x-x\\<5 ll"-«ll<£ t \]x-x\\<6 t 0 x tv x A\^ > rh m in-t f sup in-t f —f( - + )-f( ) ^—L = ttf-f] {x,u)\ . e4-0 d>0 \\x-x\\<6,0 dcf(x) = {x* € X* | {u,x*) < f\x,u) Vu € X}; if f(x) £ E we consider that dcf(x) = 0. Taking into account Theorem 2.4.14, dcf(x) ^ 0 if and only if /^(af, 0) = 0. Moreover, dcf(x) is a nonempty w*-compact subset of X* if / is Lips- chitz on a neighborhood of x. Other properties of dc are collected in the following result. Theorem 3.2.4 Let f,g : X —> R be proper functions and x £ dom/ n domg. (i) If / and g coincide on a neighborhood ofx then dcf(x) — dcg(x). (ii) If f is convex then dcf(x) = df(x). (iii) If x is a local minimum of f then 0 € dcf(x). 176 Some results and applications of convex analysis in normed spaces (iv) \imsupx_>x dcf(x) C dcf(x), where kmsupx^ dcf(x) is the set {x* G X* | 3 (xn) -+f x, 3 «) % x*, Vn G N : < G 3c/(a;n) } • (3.15) (v) If g is finite and Lipschitz on a neighborhood of x then (f + g)Hx, •) < fHx, •) + g\x, •), dcU + g)(?) c dcf(x) + dcg(x). (vi) // g is Gateaux differentiate on a neighborhood of x and Vg is con tinuous at x then (f + g)H*, •) = fHx, •) + V5(af), dc(f + g)(x) = dcf(x) + Vg(x). Proof, (i) and (iii) are obvious because p{x, •) = (flv^ix, •) for every V G N(x), while (ii) follows from the second part of Proposition 3.2.2. tnen (iv) Let x* G limsupx^ fX-dcf(y); there exist (xn) ->/ x and (a£) ^ x* such that x*n G dcf(xn) for all n G N. Let M £ I be fixed. Then (u,a;*) < p{xn,u) for every n G N, whence, by Proposition 3.2.3 (i), (u,x*) < ft(x,u), and so x* G dcf(x). (v) Let r > 0 be such that g is L-Lipschitz on B(x, r); we may as sume that L > 1. Let u £ X, e > 0 and 5 > 0 be fixed and take 6' = mm{S/{2L),r/(l + e + \\u\\)} > 0. Let v, x and t be such that \\v-u\\ (f + g)(x + tv)-(f + g)(x) f{x + tv) - f{x) g{x + tu) - g{x) t ~ t t whence, taking first the supremum with respect to v G D(u,e), we obtain that a < Ai(5) + A2(5) + Le, where . (f + g)(x + tv)-(f + g)(x) a := sup mi , ||a!-s|| U+9)(x)<(f+gm+s' Ai(S) := sup inf — —-^-L, \\x-x\\ Therefore /3 < Ai(S) + A2{5) + Le for every 5 > 0, where /3:=inf sup inf if + 9)(x+ tv) - jf + g)(X) ^ 5'>0 ||a!-»||< Taking the limit of Ai (S) + A2 (5) for S —> 0 (taking into account that A\ and A2 are nondecreasing for 5 > 0), we get inf sup inf (/ + g)(* + *0-(/ + g)(*) 5>°\\x-x\\<5,0 X tv S • f • f f( + ) ~ /(*) < inf sup inf —- - —L s>0 \\x-x\\<6,0 C(X) := {g : X -> E | g is convex and Lipschitz}; of course, €(X) is a convex cone in the vector space Rx of all functions from x X into E. Let d(X) C E be the cone generated by X*U{d[a,6] | a, b € X) x (C €{X)) and €2{X) C E be the cone generated by X*u{dfaM\a,beX}WD0, where = u conver ent Do := { ^2k>QVkdlk (Hk) C K+, ^2k>Ql*k *' ( *)*>° S } • These cones will be used below. 178 Some results and applications of convex analysis in normed spaces We call an abstract subdifferential on the nonempty class T C M. a multifunction d : X x T =4 X*, which associates to (x,f) a set denoted by df(x), satisfying condition (PI) below: (PI) 0 G limsupy_>xd/(y) + dg(x) if / € T, g € <£(X), f(x) £ 1 and a; is a local minimum of / + g, where limsup _yX df{y) is defined as in Eq. (3.15) and dg(x) is the Fenchel subdifferential of g at x. A stronger form of (PI) is (P2) 0 € ~8f(x) + dg(x) if / 6 T, g e €(X), f(x) 6 1 and a; is a local minimum of / + g. Sometimes one asks (P3) df(x) = df(x) if / e Tn€(X). Of course, (P2) holds if (P4) and (P5) below are satisfied: (P4) 0 6 df (x) if / 6 J-, f(x) € E and x is a local minimum of /, (P5) T + €(X) C T and 8{f + g)(x) C df(x) + dg(x) when / G T and 9 e €(X). Note that WX + €(X) = RX, A(X) + €(X) = A(X), T(X) + €(X) = T(X). Remark 3.2.1 From the preceding theorem we observe that Clarke's sub- differential dn and the Fenchel subdifferential d are abstract subdifferentials x on R and A(X), respectively; in fact they satisfy conditions (P1)-(P5). There are many other subdifferentials which satisfy condition (PI). Remark 3.2.2 If the abstract subdifferential d on T satisfies the stronger condition (P2) then for any proper function / 6 T one has df(x) C df(x) for all x € dom/. Indeed, if x* 6 df(x) then a; is a (local) minimum point of / + (—x*), and so, by (P2), 0 £ df(x) + d(-x*)(x) = df(x) - x*. Hence x* € df(x). The following approximate mean value theorem proves to be useful in nonconvex analysis. Convexity and monotonicity of subdifferentials 179 Theorem 3.2.5 (Zagrodny) Let (X, ||-||) be a Banach space and d be x an abstract subdifferential on T C E . Let f 6 T be Isc, a,b G X with a G dom/ and a ^ b, and r6l with r < f(b). Then there exist (xn) —>•/ c € [a, b[ and x*n G df(xn) for every n G N such that (i) r — f(a) < lim inf (6 - a, x*n), (ii) 0 < lim inf (c - xn,x*n), (iii) f^(r-/(a)) 2 Vz G U : /i(c) < h(x) + tnd[atb](x) + n~ . (3.16) Indeed, the inequality is obvious for x G [a, b] + rnUx- If x £ U \ ([a, b] + -2 r„[/x) then d[0i6](x) > r„, and so /i(x) + t„d[a)ft](a;) > 7 + inrn > ft.(c)-n . Consider i?„ — h + iu + *nd[0)(,]. Prom Eq. (3.16) we have that Hn(c) < -2 infx Hn + n ; moreover, Jf„ is Isc and bounded from below. Applying 2 _1 Corollary 1.4.2 for Hn, c, e ~ n~ and A := n , we get un G X such that _1 Hn(un) < Hn(c), ||c-u„|| < n , (3-17) 1 Hn{un) < Hn(x) + n- ||z - un|| Va; G X. (3.18) Since (un) —> c G [a, 6[, we may assume that un G intf/ for every n. 180 Some results and applications of convex analysis in normed spaces Relation (3.18) can be written as l -1 (/ + tnd[atb] + n~ dUn - x*) (u„) < (/ + tnd[aib] + n d„„ - x*) (x) for every x 6 U, where da denotes d[a,a] = ^{o}- Therefore un is a local min 1 imum of/ + (tnd[a,b] +n~ dUn -x*); the function between the parentheses being convex and Lipschitz, by (PI) we have that x 0 6 limsup <9/(2/) + d(tnd[a The formula for the Fenchel subdifferential of a sum of (continuous) con vex functions, we obtain the existence of u* g limsup^ df(y), vn € dd[atb](un) and 6* € Ux' such that u*n = x* -{tnV^ + n-Hl). Let yn £ [a, b] be such that d{a^{un) = \\un — yn\\ (such an element exists because [a, b] is compact). Because (c - un, u*) = (c - yn,v*) - (u„ - yn,v*) < - ||un - y„|| < 0. It follows that (b-a,u*n) > (b-a,x*) - ^(fr-a,^) > r - f(a) - n_11|& — a\\, whence liminf (b — a,u*) > r — /(a). _1 Similarly, {c — un,u*n) > (c — u„,x*) — n ||c-un||, whence liminf (c — un,u*n) > 0. From Eq. (3.17) we get f(un) - (un,x*) < /(c) - {c,x*), and so /(c) < liminf f(un) < limsup/(un) < /(c); hence lim/(un) = /(c) which shows that (un) —>f c. Because u*k e limsup.^^ df(u), for every k £ N there exist (u„,*) ->•/ Ufc and (u*^) ^> u£ (for n -» oo) such that u*nk G df(un>k) for all n, A; e N. Convexity and monotonicity of subdifferentials 181 It follows that ((c - uniu,u*nk)) -> (c- Ufc,u£) for n -> oo. Therefore for every A: € N there exists n'fc € N such that _1 1 1 IK,*-Wfc|| < fc , \f(un,k)- f(uk)\ < k' , \(b-a,u*nk - u*k)\ < AT , l (c-unik,u*nik)- (c-uk,u*k)\ < k~ for n > n'k. Consider (nk) C N an increasing sequence such that nk > n'k for all k e N. Let xn := u„,fc and a;* := u^ fc for nk < n < nk+i- Since for nk l \\Xn ~ C\\ < \\un>k ~ Uk\\ + \\uk - C\\ < k~ + \\uk - C\\ , l/W-Ztol^ + l/toWtol, 1 (b - a, <) = {b-a, u*k) + {b-a, < lim inf (6 — a, xn) > lim inf (b — a, u£) >r — f(a) n—>oo k—•oo and lim inf (c — xn,a;*) > lim inf (c — Ufc,u^) > 0. n—s-oo fc-+oo Taking into account that b — c = (1 — /j,)(b — a) and \\b — c|| = (1 — /i) ||b — o||, from the preceding two relations we get lim inf (b- xn,x*n) = lim inf ((1 - fi) (b- a, a:*) + {c- xn,x*n)) > (1 — /x) lim inf (b — a,xn) + lim inf (c — a;„,a;*) > K (r - /(a)). The proof is complete. • Remark 3.2.3 The conclusion of Theorem 3.2.5 remains valid if (PI) holds only for functions g in the cone £i(X) or in the cone C2PO defined on page 177. Indeed, for the first part just note that in the proof of Theorem 3.2.5 1 we used (PI) only for g = i„d[aj6j + n~ dUn — x* € £i(X). For the second part one must notice that one can find tn > 0 such that 7 + tnrn > h(c), and so Eq. (3.16) holds with d[Q>(,] replaced by d?a bi. Applying the Borwein-Preiss variational principle for p = 2, Hn, 182 Some results and applications of convex analysis in normed spaces 2 c, e := 2n~ and A := \/2/n, we find un € B(c, y/2/n) and a function 02,„ generated by the sequences (fJ.n,k)k>o C K+ with X^>0Mn,ib = 1 and (wn,fc)fc>o C B(c, \/2) (see Eq. (1.11) on page 31) such that un is a local 1 minimum of / + (tndfa M + n~ @2,n — x*). Then, in the expression of un 3 2 we have tn \\un — yn || instead of tn and 6* 6 2 / Ux* instead of &£ 6 C/x* (see also Exercise 2.40). The rest of the proof is the same. Corollary 3.2.6 Let X be a Banach space and d be an abstract sub- x differential on T C M. . If f G T is a Isc proper function then for ev ery x e dom/ there exists ((xn,xn)) C gr<9/ such that (xn) —»/ x and liminf (x — xn,x^) > 0. In particular dom/ C cl(domd/). Proof. Let x S dom/. If z is a local minimum of / then 0 belongs to limsupy^ xdf(y) by (P2) (just take g = 0). Therefore there exists (xn) —>f x and (a;*) —> 0 such that a:* 6 df(xn) for every n € N. Of course, lim (x — xn, x^) = 0 in this case, and so the conclusion holds. Assume that x is not a local minimum of /. Then there exists (xn) —> x such that f(xn) < f{x) for every n £ N. Since / is lsc, we get (xn) -»•/ x. Applying the preceding theorem for xn, x and r = f(x), there exists ((xk,n,x%tn))keN C domdf such that (xk,n) ->/ yn G [x„,a;[ for k -» oo and 0 _ < |jx-lj (/W /(^«)) < Hminf *,_>«, (a; - a;fc>„,a;^„), /(l/n) < /(*„) + |^f (/(*) - /(*»)) < /(*) for every n € N. Since \\yn - x\\ < \\xn - x\\ we have that (yn) -> x, which, together with the preceding inequality and the lower semicontinuity of /, yields (yn) ->•/ x. Then, for every n 6 N there exists k'n € N such that _1 _1 for lk*,n - Vn\\ < n , |/(a:fc,„) - f(yn)\ < n and 0 < (x - Xk,n,x*kn) every k > k'n. Let (fc„) C N be an increasing sequence such that kn > k'n for n € N. Then _1 lkfc„,n - z|| < ||a;*„,n - yn|| + \\yn - x\\ < n + \\yn - x\\, l/(**»,») - f(x)\ < \f{xkn,n) - f(yn)\ + \f(Vn) - f(x)\ < n-1 + |/(y„) - /(a:)| , 0 < (z-Z*„,„,4„,n) for all n £ N. Taking xn := Xkn,n and a;* := a;^ n, we have the desired conclusion. • Convexity and monotonicity of subdifferentials 183 When T = A(X) and d is the Fenchel subdifferential, the statement of the preceding corollary is very close to assertion (i) of Theorem 3.1.2. Theorem 3.2.7 Let X be a Banach space, d be an abstract subdifferential X on T C K and f £ T be a Isc proper function. If df is a monotone multi function then f is convex. Proof. We prove the theorem in three steps: 1) a £ dom/, b £ domdf => [a,b] C dom/, 2) grd/ C gr<9/, 3) / is convex. 1) Assume that there exists b' £ [a, b] \ dom /; so b' — Xa + (1 — X)b with A £ ]0,1[. Fix y* £ df(b) and take r £ ffi, 1 r>/(a) + A- (l-A)||6-a||.||2/*||. By Theorem 3.2.5, there exist (xn) -»/ c £ [a, b'[ and x* € df(xn) for every n € N such that liminf (&' — a;n,a;*) > 0, liminf (b' — a, a;*) > r — f(a). Since df is monotone we get the contradiction l|6-a||-||y*ll>ll&-c||-||y*||><6-c)i/*> = liminf (b — xn,y*) > liminf (b — xn,x*n) 1 - liminf (A(l - A)" (b' - a,x*n) + (b' - xn, x*n)) -1 > A(l - A) liminf (6-a,x*) + liminf (b' -xn,x*n) >A(1_A)-i(r_/(a)). 2) Let x* 6 df{b) with b 6 X. Consider x £ dom/. Applying again Theorem 3.2.5 for r = f(b), there exist (xn) ->f c € [x, b[ and x* € df(xn) for every n 6 N such that liminf (b-xn,x*n) > gff} (/(6) - /(*)). Since 9/ is monotone and c = Ax + (1 — A)6 with A € ]0,1], we have that A (6 — x,x*) = (b — c,x*) = liminf (& — xn,x*) > liminf (b — xn,a;*) >A(/(6)-/(x)), and so (x — b, x*) < f(x) — f(b) for every x £ dom/. Therefore x* £ df(x). 3) Let x, y £ dom / and z := (1 — X)x + Xy with A £ ]0,1[. By Corollary 3.2.6, there exists (yn) c dom df such that (yn) ->•/ y. Let a;n := (1-A)x + Ay„ for n £ N; by 1) we have that zn £ dom/ for every n. Using again 184 Some results and applications of convex analysis in normed spaces c TO sucri tnat z — z Corollary 3.2.6, there exists ((zn,k,zn k)) S f ( n,k) >•/ n — z z for k —> oo and liminffc_>oo(^n n,ki nk) — 0. By 2) we have that z*nk G df(zn>k), and so (x-zntk,z^k)+f(zn,k) < f(x), (yn- zn,k,znk) + f{zn>k) < f(yn) for all n,k G N. Multiplying the first inequality by (1 — A) > 0 and the second one by A > 0 we get (zn - zn,k, z*n>k) + f(zn,k) < (1 - A)/(x) + Xf(yn) Vn, k G N. Taking the limit inferior for k —> oo we get f(zn) < (1 — A)/(x) + Xf(yn) for every n. Passing to the limit inferior for n —> oo and taking into account the lower semicontinuity of / and the fact that (yn) ->/ y we get f(z) < (1 — A)/(a;) + Xf(y), and so / is convex. • We give now another short proof for the Rockafellar theorem on the maximal monotonicity of the subdifferential of a lsc convex function. Theorem 3.2.8 Let X be a Banach space and f € T(X). Then df is maximal monotone. Proof. Let x G X and x* ^ df(x). Then x is not a minimum point for g := f — x*, and so there exists z E X such that g{z) < g(x). Let r G E be between these numbers. By Theorem 3.2.5 applied for d on c r r(X), there exists ((yn,yn)) S 9g such that (yn) —>fl y € [z,x[ and lim inf (x - yn,y*n) > |f5|[ (r - g{z)) > 0. Therefore (x - yn,y*n) > 0 for n > no, for some n0 G N. It follows that y* := yno + x* € df(y), with V '•= 2/n0) and (x — y,x* — y*) < 0. Therefore df is maximal monotone. • Theorem 3.2.5 can also be used for proving that two convex functions with the same subdifferential differ by an additive constant. In fact one obtains even more. Theorem 3.2.9 Let C be an open convex subset of the Banach space X jf and d an abstract subdifferential on T C IK . Let f G T be proper and lower semicontinuous and g G T(X). Assume there exists e G R+ such that df(x) C dg{x) + eUx* Vi G C. (3.19) Then C D dom / = C f) dom g and 9{x) - g{y) ~e\\x- y\\ < f(x) - f(y) < g(x) - g(y) +e\\x~ y\\ (3.20) Convexity and monotonicity of subdifferentials 185 for all x £ C and y £ C fl domg. Proof. Consider 7 > e. By Corollary 3.2.6 we have that dom/ C cl(dom9/), and so C fl domdf ^ 0. Let y £ C fl dom9/ and x £ C. Prom our assumption we have that y £ doming, and so y € domg. We are going to prove that /(*) - /(!/) < $(*) - 5(2/) + e II* - 2/11 • (3-21) The inequality is obvious if x £ domg or 1 = y, and so assume that x £ domg and x ^ y. Because dg(y) ^ 0, we have that g'(y,u) > -00 for all u £ X. Consider ip : E -»E defined by lim g'(y + tu,u) = lim and this quantity is finite. Therefore there exists to £ ]0, ||:r — y||[ such that g'(y + t0u,u) (-r ,j/*) G (grC)*). Since \\T*y*\\ > p3 > p3 \\x*\\, (ii) does not hold for pi= p3. (iv) =^ (iii) when T is onto. Since T is onto and Z, Y are Banach spaces, • Y'; the operator H is Q-convex if
f(,Vi) < /(2/2)• For such a function we consider that /(oo) = +00. It is obvious that every function is {0}-increasing, and that a linear functional ip : Y —> E is Q-increasing if and only if (p(y) > 0 for every y £ Q. One defines similarly the Q-decreasing functions. In the next result we mention some methods for deriving new convex functions from known ones. For a, (3 £ E we set a V /? := max{a,/?} and a A P := min{a, /3}.
dom/ ^ 0 and 3x* G X*, a G K, Vrr G X : f(x) > (x,x*) +a; (ii) h* G r(X) dom/i / 8 and 3i 6 I, a e E, Va:* £ X* : /i(:r*) > (x,a;*) + a. • The following result is fundamental in duality theory. Ax = y, A*j/* G fy/fr). Lagrange multiplier of problem (P). Note that when H is finite-valued (i.e. dom# = X) and continuous,