Convex Analysis1

Total Page:16

File Type:pdf, Size:1020Kb

Convex Analysis1 Convex Analysis1 Patrick Cheridito Princeton University, Spring 2013 1Many thanks to Andreas Hamel for providing his lecture notes. Large parts of chapters 2-4 are based on them. 2 Contents Contents 2 1 Convex Analysis in Rd 5 1.1 Subspaces, affine sets, convex sets, cones and half-spaces . 5 1.2 Separation results in finite dimensions . 8 1.3 Linear, affine and convex functions . 12 1.4 Derivatives, directional derivatives and sub-gradients . 16 1.5 Convex conjugates . 19 1.6 Inf-convolution . 23 2 General Vector Spaces 27 2.1 Definitions . 27 2.2 Zorn's lemma and extension results . 30 2.3 Algebraic interior and separation results . 33 2.4 Directional derivatives and sub-gradients . 35 3 Topological Vector Spaces 37 3.1 Topological spaces . 37 3.2 Continuous linear functionals and extension results . 42 3.3 Separation with continuous linear functionals . 44 3.4 Continuity of convex functions . 47 3.5 Derivatives and sub-gradients . 48 3.6 Dual pairs . 50 3.7 Convex conjugates . 53 3.8 Inf-convolution . 55 4 Convex Optimization 59 4.1 Perturbation and the dual problem . 59 4.2 Lagrangians and saddle points . 65 4.3 Karush{Kuhn{Tucker-type conditions . 67 3 4 CONTENTS Chapter 1 d Convex Analysis in R The following notation is used: • d 2 N := f1; 2;:::g d • ei is the i-th unit vector in R Pd d • hx; yi := i=1 xiyi for x; y 2 R p • jjxjj := hx; xi for x 2 Rd d • B"(x) := y 2 R : jjx − yjj ≤ " • R+ := fx 2 R : x ≥ 0g, R++ := fx 2 R : x > 0g • x _ y := max fx; yg and x ^ y := min fx; yg for x; y 2 R 1.1 Subspaces, affine sets, convex sets, cones and half-spaces Definition 1.1.1 Let C be a subset of Rd. C is a subspace of Rd if λx + y 2 C for all x; y 2 C and λ 2 R: C is an affine set if λx + (1 − λ)y 2 C for all x; y 2 C and λ 2 R: C is a convex set if λx + (1 − λ)y 2 C for all x; y 2 C and λ 2 [0; 1]: C is a cone if λx 2 C for all x 2 C and λ 2 R++: 5 6 CHAPTER 1. CONVEX ANALYSIS IN RD Exercise 1.1.2 Let C; D be non-empty subsets of Rd. 1. Show that if C; D are subspaces, then so is C − D := fx − y : x 2 C; y 2 Dg ; and the same is true for affine sets, convex sets and cones. 2. Show that if C is affine, then C + v is affine for every v 2 Rk. 3. Show that if C is affine and contains 0, it is a subspace. 4. Show that if C is affine and v 2 C, then C − v = C − C is a subspace. 5. Show that the intersection of arbitrarily many subspaces is a subspace, and that the same is true for affine subsets, convex subsets and cones. 6. Show that there exists a smallest subspace containing C, and that the same is true for affine sets, convex sets and cones. Definition 1.1.3 If C is a non-empty subset of Rd, we denote by lin C, aff C, conv C, cone C the smallest subspace, affine set, convex set, cone containing C, respectively. Exercise 1.1.4 Let C be a non-empty subset of Rd. Show that ( n ) X lin C = λixi : n 2 N; λi 2 R; xi 2 C i=1 ( n n ) X X aff C = λixi : n 2 N; λi 2 R; xi 2 C; λi = 1 i=1 i=1 ( n n ) X X conv C = λixi : n 2 N; λi 2 R+; xi 2 C; λi = 1 i=1 i=1 cone C = fλx : λ 2 R++; x 2 Cg Definition 1.1.5 The dimension of an affine subset M of Rd is the dimension of the subspace M − M. The dimension of an arbitrary subset C is the dimension of aff C. Definition 1.1.6 Let C be a non-empty subset of Rd. The dual cone of C is the set ∗ d C := z 2 R : hx; zi ≥ 0 for all x 2 C : Exercise 1.1.7 Show that the dual cone C∗ of a non-empty subset C ⊆ Rd is a closed convex cone and C is contained in C∗∗. Definition 1.1.8 The recession cone 0+C of a subset C of Rd consists of all y 2 R satisfying x + λy 2 C for all x 2 C and λ 2 R++: Every y 2 0+C n f0g is called a direction of recession for C. 1.1. SUBSPACES, AFFINE SETS, CONVEX SETS, CONES AND HALF-SPACES7 Definition 1.1.9 Let C be a subset of Rd. The closure cl C of C is the smallest closed subset of Rd containing C. The interior int C consists of all x 2 C such that B"(x) ⊆ C for some " 2 R++. The relative interior ri C is the set of all x 2 C such that B"(x) \ aff C ⊆ C for some " 2 R++. The boundary of C is the set bd C := cl C n int C. The relative boundary is rbd C := cl C n ri C Exercise 1.1.10 1. Show that an affine subset of Rd is closed. 2. Show that the closure of a cone is a cone. 3. Show that the closure of a convex set is convex. Lemma 1.1.11 Let C be a non-empty convex subset of Rd and λ 2 (0; 1]. If int C 6= ;, then λint C + (1 − λ)cl C ⊆ int C: (1.1.1) If ri C 6= ;, then λri C + (1 − λ)cl C ⊆ ri C (1.1.2) In particular, int C and ri C are convex. Proof. Let x 2 int C, y 2 cl C and λ 2 (0; 1]. There exists " > 0 such that B2"(x) ⊆ C and z 2 C such that (1 − λ)jjy − zjj ≤ λε. Choose v 2 Bλε(0). Then v 1 − λ w = + (y − z) 2 B (0); λ λ 2" and therefore, λx + (1 − λ)y + v = λ(x + w) + (1 − λ)z 2 C: d This shows (1.1.1). (1.1.2) follows by working in aff C instead of R . Lemma 1.1.12 Let C be a convex subset of Rd. Then int C 6= ; if and only if aff C = Rd. Proof. If x 2 int C, then 0 2 int C − x, and it follows that d aff (C) − x = aff (C − x) = lin (C − x) = R : On the other hand, if aff C = Rd, choose x 2 C. Then d lin (C − x) = aff (C − x) = aff (C) − x = R : So there there exist d vectors x1; : : : ; xd in C such that vi := xi − x are linearly independent. Since C is convex, one has 1 1 (x + x + ··· + x ) + λv 2 C for jλj ≤ and i = 1; : : : ; d; d + 1 1 d i d + 1 8 CHAPTER 1. CONVEX ANALYSIS IN RD and therefore, 1 (x + x + ··· + x ) + V ⊆ C; d + 1 1 d nPd Pd 1 o Pd where V := i=1 λivi : i=1 jλij ≤ d+1 . So since n(x) := i=1 jλij for x = Pd d i=1 λivi defines a norm and all norms on R are equivalent, there exists an " > 0 such that 1 (x + x + ··· + x ) + B (0) ⊆ C: d + 1 1 d " Corollary 1.1.13 Let C be a non-empty convex subset of Rd. Then ri C is dense in C. In particular, ri C is non-empty. Proof. If C consists of only one point x0, then ri C = C = fx0g. If C contains at least two different points, one can, by shifting, assume that one of them is 0. Then lin C = aff C is at least one-dimensional. So by restricting to lin C, one can assume that lin C = Rd. It follows from Lemma 1.1.12 that ri C 6= ;. Now the corollary follows from Lemma 1.1.11. Definition 1.1.14 A half-space in Rd is a set of the form d d x 2 R : hx; zi ≥ c for some z 2 R n f0g and c 2 R: d d We say a subset C of R is supported at x0 2 C by z 2 R n f0g if hx0; zi = infx2C hx; zi. d d Note that if a subset C of R is supported at x0 2 C by some z 2 R n f0g, then x0 is in the boundary of C and C is contained in the half-space d x 2 R : hx; zi ≥ hx0; zi : 1.2 Separation results in finite dimensions d Lemma 1.2.1 Let C be a non-empty closed subset of R . Then there exists x0 2 C such that jjx0jj = inf jjxjj: x2C If in addition, C is convex, then x0 is unique. Proof. For fixed y 2 C, the set fx 2 C : jjxjj ≤ jjyjjg is closed and bounded. So the existence of x0 follows because the norm is continuous. If C is convex and x0; x1 are two different norm minimizers, one has x + x jj 0 1 jj < jjx jj = jjx jj; 2 0 1 a contradiction.
Recommended publications
  • On the Cone of a Finite Dimensional Compact Convex Set at a Point
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector On the Cone of a Finite Dimensional Compact Convex Set at a Point C. H. Sung Department of Mathematics University of Califmia at San Diego La Joll41, California 92093 and Department of Mathematics San Diego State University San Diego, California 92182 and B. S. Tam* Department of Mathematics Tamkang University Taipei, Taiwan 25137, Republic of China Submitted by Robert C. Thompson ABSTRACT Four equivalent conditions for a convex cone in a Euclidean space to be an F,-set are given. Our result answers in the negative a recent open problem posed by Tam [5], characterizes the barrier cone of a convex set, and also provides an alternative proof for the known characterizations of the inner aperture of a convex set as given by BrBnsted [2] and Larman [3]. 1. INTRODUCTION Let V be a Euclidean space and C be a nonempty convex set in V. For each y E C, the cone of C at y, denoted by cone(y, C), is the cone *The work of this author was supported in part by the National Science Council of the Republic of China. LINEAR ALGEBRA AND ITS APPLICATIONS 90:47-55 (1987) 47 0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017 00243795/87/$3.50 48 C. H. SUNG AND B. S. TAM { a(r - Y) : o > 0 and x E C} (informally called a point’s cone). On 23 August 1983, at the International Congress of Mathematicians in Warsaw, in a short communication session, the second author presented the paper [5] and posed the following open question: Is it true that, given any cone K in V, there always exists a compact convex set C whose point’s cone at some given point y is equal to K? In view of the relation cone(y, C) = cone(0, C - { y }), the given point y may be taken to be the origin of the space.
    [Show full text]
  • 1 Overview 2 Elements of Convex Analysis
    AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 2 | Wednesday, January 27th 1 Overview In our previous lecture we discussed several applications of optimization, introduced basic terminol- ogy, and proved Weierstass' theorem (circa 1830) which gives a sufficient condition for existence of an optimal solution to an optimization problem. Today we will discuss basic definitions and prop- erties of convex sets and convex functions. We will then prove the separating hyperplane theorem and see an application of separating hyperplanes in machine learning. We'll conclude by describing the seminal perceptron algorithm (1957) which is designed to find separating hyperplanes. 2 Elements of Convex Analysis We will primarily consider optimization problems over convex sets { sets for which any two points are connected by a line. We illustrate some convex and non-convex sets in Figure 1. Definition. A set S is called a convex set if any two points in S contain their line, i.e. for any x1; x2 2 S we have that λx1 + (1 − λ)x2 2 S for any λ 2 [0; 1]. In the previous lecture we saw linear regression an example of an optimization problem, and men- tioned that the RSS function has a convex shape. We can now define this concept formally. n Definition. For a convex set S ⊆ R , we say that a function f : S ! R is: • convex on S if for any two points x1; x2 2 S and any λ 2 [0; 1] we have that: f (λx1 + (1 − λ)x2) ≤ λf(x1) + 1 − λ f(x2): • strictly convex on S if for any two points x1; x2 2 S and any λ 2 [0; 1] we have that: f (λx1 + (1 − λ)x2) < λf(x1) + (1 − λ) f(x2): We illustrate a convex function in Figure 2.
    [Show full text]
  • CORE View Metadata, Citation and Similar Papers at Core.Ac.Uk
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Bulgarian Digital Mathematics Library at IMI-BAS Serdica Math. J. 27 (2001), 203-218 FIRST ORDER CHARACTERIZATIONS OF PSEUDOCONVEX FUNCTIONS Vsevolod Ivanov Ivanov Communicated by A. L. Dontchev Abstract. First order characterizations of pseudoconvex functions are investigated in terms of generalized directional derivatives. A connection with the invexity is analysed. Well-known first order characterizations of the solution sets of pseudolinear programs are generalized to the case of pseudoconvex programs. The concepts of pseudoconvexity and invexity do not depend on a single definition of the generalized directional derivative. 1. Introduction. Three characterizations of pseudoconvex functions are considered in this paper. The first is new. It is well-known that each pseudo- convex function is invex. Then the following question arises: what is the type of 2000 Mathematics Subject Classification: 26B25, 90C26, 26E15. Key words: Generalized convexity, nonsmooth function, generalized directional derivative, pseudoconvex function, quasiconvex function, invex function, nonsmooth optimization, solution sets, pseudomonotone generalized directional derivative. 204 Vsevolod Ivanov Ivanov the function η from the definition of invexity, when the invex function is pseudo- convex. This question is considered in Section 3, and a first order necessary and sufficient condition for pseudoconvexity of a function is given there. It is shown that the class of strongly pseudoconvex functions, considered by Weir [25], coin- cides with pseudoconvex ones. The main result of Section 3 is applied to characterize the solution set of a nonlinear programming problem in Section 4. The base results of Jeyakumar and Yang in the paper [13] are generalized there to the case, when the function is pseudoconvex.
    [Show full text]
  • The Hahn-Banach Theorem and Infima of Convex Functions
    The Hahn-Banach theorem and infima of convex functions Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa 923-1292, Japan first CORE meeting, LMU Munich, 7 May, 2016 Normed spaces Definition A normed space is a linear space E equipped with a norm k · k : E ! R such that I kxk = 0 $ x = 0, I kaxk = jajkxk, I kx + yk ≤ kxk + kyk, for each x; y 2 E and a 2 R. Note that a normed space E is a metric space with the metric d(x; y) = kx − yk: Definition A Banach space is a normed space which is complete with respect to the metric. Examples For 1 ≤ p < 1, let p N P1 p l = f(xn) 2 R j n=0 jxnj < 1g and define a norm by P1 p 1=p k(xn)k = ( n=0 jxnj ) : Then lp is a (separable) Banach space. Examples Classically the normed space 1 N l = f(xn) 2 R j (xn) is boundedg with the norm k(xn)k = sup jxnj n is an inseparable Banach space. However, constructively, it is not a normed space. Linear mappings Definition A mapping T between linear spaces E and F is linear if I T (ax) = aTx, I T (x + y) = Tx + Ty for each x; y 2 E and a 2 R. Definition A linear functional f on a linear space E is a linear mapping from E into R. Bounded linear mappings Definition A linear mapping T between normed spaces E and F is bounded if there exists c ≥ 0 such that kTxk ≤ ckxk for each x 2 E.
    [Show full text]
  • Sharp Finiteness Principles for Lipschitz Selections: Long Version
    Sharp finiteness principles for Lipschitz selections: long version By Charles Fefferman Department of Mathematics, Princeton University, Fine Hall Washington Road, Princeton, NJ 08544, USA e-mail: [email protected] and Pavel Shvartsman Department of Mathematics, Technion - Israel Institute of Technology, 32000 Haifa, Israel e-mail: [email protected] Abstract Let (M; ρ) be a metric space and let Y be a Banach space. Given a positive integer m, let F be a set-valued mapping from M into the family of all compact convex subsets of Y of dimension at most m. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of F with the sharp value of the finiteness number. Contents 1. Introduction. 2 1.1. Main definitions and main results. 2 1.2. Main ideas of our approach. 3 2. Nagata condition and Whitney partitions on metric spaces. 6 arXiv:1708.00811v2 [math.FA] 21 Oct 2017 2.1. Metric trees and Nagata condition. 6 2.2. Whitney Partitions. 8 2.3. Patching Lemma. 12 3. Basic Convex Sets, Labels and Bases. 17 3.1. Main properties of Basic Convex Sets. 17 3.2. Statement of the Finiteness Theorem for bounded Nagata Dimension. 20 Math Subject Classification 46E35 Key Words and Phrases Set-valued mapping, Lipschitz selection, metric tree, Helly’s theorem, Nagata dimension, Whitney partition, Steiner-type point. This research was supported by Grant No 2014055 from the United States-Israel Binational Science Foundation (BSF). The first author was also supported in part by NSF grant DMS-1265524 and AFOSR grant FA9550-12-1-0425.
    [Show full text]
  • Applications of Convex Analysis Within Mathematics
    Noname manuscript No. (will be inserted by the editor) Applications of Convex Analysis within Mathematics Francisco J. Arag´onArtacho · Jonathan M. Borwein · Victoria Mart´ın-M´arquez · Liangjin Yao July 19, 2013 Abstract In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis. Keywords Adjoint · Asplund averaging · autoconjugate representer · Banach limit · Chebyshev set · convex functions · Fenchel duality · Fenchel conjugate · Fitzpatrick function · Hahn{Banach extension theorem · infimal convolution · linear relation · Minty surjectivity theorem · maximally monotone operator · monotone operator · Moreau's decomposition · Moreau envelope · Moreau's max formula · Moreau{Rockafellar duality · normal cone operator · renorming · resolvent · Sandwich theorem · subdifferential operator · sum theorem · Yosida approximation Mathematics Subject Classification (2000) Primary 47N10 · 90C25; Secondary 47H05 · 47A06 · 47B65 Francisco J. Arag´onArtacho Centre for Computer Assisted Research Mathematics and its Applications (CARMA),
    [Show full text]
  • On Order Preserving and Order Reversing Mappings Defined on Cones of Convex Functions MANUSCRIPT
    ON ORDER PRESERVING AND ORDER REVERSING MAPPINGS DEFINED ON CONES OF CONVEX FUNCTIONS MANUSCRIPT Lixin Cheng∗ Sijie Luo School of Mathematical Sciences Yau Mathematical Sciences Center Xiamen University Tsinghua University Xiamen 361005, China Beijing 100084, China [email protected] [email protected] ABSTRACT In this paper, we first show that for a Banach space X there is a fully order reversing mapping T from conv(X) (the cone of all extended real-valued lower semicontinuous proper convex functions defined on X) onto itself if and only if X is reflexive and linearly isomorphic to its dual X∗. Then we further prove the following generalized “Artstein-Avidan-Milman” representation theorem: For every fully order reversing mapping T : conv(X) → conv(X) there exist a linear isomorphism ∗ ∗ ∗ U : X → X , x0, ϕ0 ∈ X , α> 0 and r0 ∈ R so that ∗ (Tf)(x)= α(Ff)(Ux + x0)+ hϕ0, xi + r0, ∀x ∈ X, where F : conv(X) → conv(X∗) is the Fenchel transform. Hence, these resolve two open questions. We also show several representation theorems of fully order preserving mappings de- fined on certain cones of convex functions. For example, for every fully order preserving mapping S : semn(X) → semn(X) there is a linear isomorphism U : X → X so that (Sf)(x)= f(Ux), ∀f ∈ semn(X), x ∈ X, where semn(X) is the cone of all lower semicontinuous seminorms on X. Keywords Fenchel transform · order preserving mapping · order reversing mapping · convex function · Banach space 1 Introduction An elegant theorem of Artstein-Avidan and Milman [5] states that every fully order reversing (resp.
    [Show full text]
  • Ce Document Est Le Fruit D'un Long Travail Approuvé Par Le Jury De Soutenance Et Mis À Disposition De L'ensemble De La Communauté Universitaire Élargie
    AVERTISSEMENT Ce document est le fruit d'un long travail approuvé par le jury de soutenance et mis à disposition de l'ensemble de la communauté universitaire élargie. Il est soumis à la propriété intellectuelle de l'auteur. Ceci implique une obligation de citation et de référencement lors de l’utilisation de ce document. D'autre part, toute contrefaçon, plagiat, reproduction illicite encourt une poursuite pénale. Contact : [email protected] LIENS Code de la Propriété Intellectuelle. articles L 122. 4 Code de la Propriété Intellectuelle. articles L 335.2- L 335.10 http://www.cfcopies.com/V2/leg/leg_droi.php http://www.culture.gouv.fr/culture/infos-pratiques/droits/protection.htm THESE` pr´esent´eepar NGUYEN Manh Cuong en vue de l'obtention du grade de DOCTEUR DE L'UNIVERSITE´ DE LORRAINE (arr^et´eminist´eriel du 7 Ao^ut 2006) Sp´ecialit´e: INFORMATIQUE LA PROGRAMMATION DC ET DCA POUR CERTAINES CLASSES DE PROBLEMES` EN APPRENTISSAGE ET FOUILLE DE DONEES.´ Soutenue le 19 mai 2014 devant le jury compos´ede Rapporteur BENNANI Youn`es Professeur, Universit´eParis 13 Rapporteur RAKOTOMAMONJY Alain Professeur, Universit´ede Rouen Examinateur GUERMEUR Yann Directeur de recherche, LORIA-Nancy Examinateur HEIN Matthias Professeur, Universit´eSaarland Examinateur PHAM DINH Tao Professeur ´em´erite, INSA-Rouen Directrice de th`ese LE THI Hoai An Professeur, Universit´ede Lorraine Co-encadrant CONAN-GUEZ Brieuc MCF, Universit´ede Lorraine These` prepar´ ee´ a` l'Universite´ de Lorraine, Metz, France au sein de laboratoire LITA Remerciements Cette th`ese a ´et´epr´epar´ee au sein du Laboratoire d'Informatique Th´eorique et Appliqu´ee (LITA) de l'Universit´ede Lorraine - France, sous la co-direction du Madame le Professeur LE THI Hoai An, Directrice du laboratoire Laboratoire d'Informatique Th´eorique et Ap- pliqu´ee(LITA), Universit´ede Lorraine et Ma^ıtre de conf´erence CONAN-GUEZ Brieuc, Institut Universitaire de Technologie (IUT), Universit´ede Lorraine, Metz.
    [Show full text]
  • The Continuity of Additive and Convex Functions, Which Are Upper Bounded
    THE CONTINUITY OF ADDITIVE AND CONVEX FUNCTIONS, WHICH ARE UPPER BOUNDED ON NON-FLAT CONTINUA IN Rn TARAS BANAKH, ELIZA JABLO NSKA,´ WOJCIECH JABLO NSKI´ Abstract. We prove that for a continuum K ⊂ Rn the sum K+n of n copies of K has non-empty interior in Rn if and only if K is not flat in the sense that the affine hull of K coincides with Rn. Moreover, if K is locally connected and each non-empty open subset of K is not flat, then for any (analytic) non-meager subset A ⊂ K the sum A+n of n copies of A is not meager in Rn (and then the sum A+2n of 2n copies of the analytic set A has non-empty interior in Rn and the set (A − A)+n is a neighborhood of zero in Rn). This implies that a mid-convex function f : D → R, defined on an open convex subset D ⊂ Rn is continuous if it is upper bounded on some non-flat continuum in D or on a non-meager analytic subset of a locally connected nowhere flat subset of D. Let X be a linear topological space over the field of real numbers. A function f : X → R is called additive if f(x + y)= f(x)+ f(y) for all x, y ∈ X. R x+y f(x)+f(y) A function f : D → defined on a convex subset D ⊂ X is called mid-convex if f 2 ≤ 2 for all x, y ∈ D. Many classical results concerning additive or mid-convex functions state that the boundedness of such functions on “sufficiently large” sets implies their continuity.
    [Show full text]
  • Lipschitz Continuity of Convex Functions
    Lipschitz Continuity of Convex Functions Bao Tran Nguyen∗ Pham Duy Khanh† November 13, 2019 Abstract We provide some necessary and sufficient conditions for a proper lower semi- continuous convex function, defined on a real Banach space, to be locally or globally Lipschitz continuous. Our criteria rely on the existence of a bounded selection of the subdifferential mapping and the intersections of the subdifferential mapping and the normal cone operator to the domain of the given function. Moreover, we also point out that the Lipschitz continuity of the given function on an open and bounded (not necessarily convex) set can be characterized via the existence of a bounded selection of the subdifferential mapping on the boundary of the given set and as a consequence it is equivalent to the local Lipschitz continuity at every point on the boundary of that set. Our results are applied to extend a Lipschitz and convex function to the whole space and to study the Lipschitz continuity of its Moreau envelope functions. Keywords Convex function, Lipschitz continuity, Calmness, Subdifferential, Normal cone, Moreau envelope function. Mathematics Subject Classification (2010) 26A16, 46N10, 52A41 1 Introduction arXiv:1911.04886v1 [math.FA] 12 Nov 2019 Lipschitz continuous and convex functions play a significant role in convex and nonsmooth analysis. It is well-known that if the domain of a proper lower semicontinuous convex function defined on a real Banach space has a nonempty interior then the function is continuous over the interior of its domain [3, Proposition 2.111] and as a consequence, it is subdifferentiable (its subdifferential is a nonempty set) and locally Lipschitz continuous at every point in the interior of its domain [3, Proposition 2.107].
    [Show full text]
  • Nonlinear Monotone Operators with Values in 9(X, Y)
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 140, 83-94 (1989) Nonlinear Monotone Operators with Values in 9(X, Y) N. HADJISAVVAS, D. KRAVVARITIS, G. PANTELIDIS, AND I. POLYRAKIS Department of Mathematics, National Technical University of Athens, Zografou Campus, 15773 Athens, Greece Submitted by C. Foias Received April 2, 1987 1. INTRODUCTION Let X be a topological vector space, Y an ordered topological vector space, and 9(X, Y) the space of all linear and continuous mappings from X into Y. If T is an operator from X into 9(X, Y) (generally multivalued) with domain D(T), T is said to be monotone if for all xieD(T) and Aie T(x,), i= 1, 2. In the scalar case Y = R, this definition coincides with the well known definition of monotone operator (cf. [ 12,4]). An important subclass of monotone operators T: X+ 9(X, Y) consists of the subdifferentials of convex operators from X into Y, which have been studied by Valadier [ 171, Kusraev and Kutateladze [ 111, Papageorgiou [ 131, and others. Some properties of monotone operators have been investigated by Kirov [7, 81, in connection with the study of the differentiability of convex operators. In this paper we begin our investigation of the properties of monotone operators from X into 3(X, Y) by introducing, in Section 3, the notion of hereditary order convexity of subsets of 9(X, Y). This notion plays a key role in our discussion and expresses a kind of separability between sets and points, Its interest relies on the fact that the images of maximal monotone operators are hereditarily order-convex.
    [Show full text]
  • ABOUT STATIONARITY and REGULARITY in VARIATIONAL ANALYSIS 1. Introduction the Paper Investigates Extremality, Stationarity and R
    1 ABOUT STATIONARITY AND REGULARITY IN VARIATIONAL ANALYSIS 2 ALEXANDER Y. KRUGER To Boris Mordukhovich on his 60th birthday Abstract. Stationarity and regularity concepts for the three typical for variational analysis classes of objects { real-valued functions, collections of sets, and multifunctions { are investi- gated. An attempt is maid to present a classification scheme for such concepts and to show that properties introduced for objects from different classes can be treated in a similar way. Furthermore, in many cases the corresponding properties appear to be in a sense equivalent. The properties are defined in terms of certain constants which in the case of regularity proper- ties provide also some quantitative characterizations of these properties. The relations between different constants and properties are discussed. An important feature of the new variational techniques is that they can handle nonsmooth 3 functions, sets and multifunctions equally well Borwein and Zhu [8] 4 1. Introduction 5 The paper investigates extremality, stationarity and regularity properties of real-valued func- 6 tions, collections of sets, and multifunctions attempting at developing a unifying scheme for defining 7 and using such properties. 8 Under different names this type of properties have been explored for centuries. A classical 9 example of a stationarity condition is given by the Fermat theorem on local minima and max- 10 ima of differentiable functions. In a sense, any necessary optimality (extremality) conditions de- 11 fine/characterize certain stationarity (singularity/irregularity) properties. The separation theorem 12 also characterizes a kind of extremal (stationary) behavior of convex sets. 13 Surjectivity of a linear continuous mapping in the Banach open mapping theorem (and its 14 extension to nonlinear mappings known as Lyusternik-Graves theorem) is an example of a regularity 15 condition.
    [Show full text]