DOCSLIB.ORG

Lecture 13 Introduction to Quasiconvex Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 13 Introduction to Quasiconvex Analysis Lecture 13 Introduction to quasiconvex analysis Didier Aussel Univ. de Perpignan – p.1/48 Outline of lecture 13 I- Introduction II- Normal approach a- First definitions b- Adjusted sublevel sets and normal operator III- Quasiconvex optimization a- Optimality conditions b- Convex constraint case c- Nonconvex constraint case – p.2/48 Quasiconvexity A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if, for all x, y ∈ K and all t ∈ [0, 1], f(tx + (1 − t)y) ≤ max{f(x),f(y)}. – p.3/48 Quasiconvexity A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if, for all λ ∈ IR, the sublevel set Sλ = {x ∈ X : f(x) ≤ λ} is convex. – p.3/48 Quasiconvexity A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if, for all λ ∈ IR, the sublevel set Sλ = {x ∈ X : f(x) ≤ λ} is convex. A function f : X → IR ∪ {+∞} is said to be semistrictly quasiconvex on K if, f is quasiconvex and for any x,y ∈ K, f(x) <f(y) ⇒ f(z) <f(y), ∀ z ∈ [x,y[. – p.3/48 I Introduction – p.4/48 f differentiable f is quasiconvex iff df is quasimonotone iff df(x)(y − x) > 0 ⇒ df(y)(y − x) ≥ 0 f is quasiconvex iff ∂f is quasimonotone iff ∃ x∗ ∈ ∂f(x) : hx∗,y − xi > 0 ⇒∀ y∗ ∈ ∂f(y), hy∗,y − xi≥ 0 – p.5/48 Why not a subdifferential for quasiconvex programming? – p.6/48 Why not a subdifferential for quasiconvex programming? No (upper) semicontinuity of ∂f if f is not supposed to be Lipschitz – p.6/48 Why not a subdifferential for quasiconvex programming? No (upper) semicontinuity of ∂f if f is not supposed to be Lipschitz No sufficient optimality condition HH¨ x¯ ∈ Sstr(∂f,C)=¨¨⇒H x¯ ∈ arg min f C – p.6/48 II Normal approach of quasiconvex analysis – p.7/48 II Normal approach a- First definitions – p.8/48 A first approach Sublevel set: Sλ = {x ∈ X : f(x) ≤ λ} > Sλ = {x ∈ X : f(x) < λ} Normal operator: X∗ Define Nf (x) : X → 2 by Nf (x) = N(Sf(x),x) ∗ ∗ ∗ = {x ∈ X : hx ,y − xi≤ 0, ∀ y ∈ Sf(x)}. > With the corresponding definition for Nf (x) – p.9/48 But ... Nf (x) = N(Sf(x),x) has no upper-semicontinuity properties > > Nf (x) = N(Sf(x),x) has no quasimonotonicity properties – p.10/48 But ... Nf (x) = N(Sf(x),x) has no upper-semicontinuity properties > > Nf (x) = N(Sf(x),x) has no quasimonotonicity properties Example Define f : R2 → R by 8 |a| + |b| , if |a| + |b|≤ 1 f (a,b)= < . 1, if |a| + |b| > 1 : Then f is quasiconvex. Consider x = (10, 0), x∗ = (1, 2), y = (0, 10) and y∗ = (2, 1). We see that x∗ ∈ N <(x) and y∗ ∈ N < (y) (since |a| + |b| < 1 implies (1, 2) · (a − 10,b) ≤ 0 and (2, 1) · (a,b − 10) ≤ 0) while hx∗, y − xi > 0 and hy∗, y − xi < 0. Hence N < is not quasimonotone. – p.10/48 But ... Nf (x) = N(Sf(x),x) has no upper-semicontinuity properties > > Nf (x) = N(Sf(x),x) has no quasimonotonicity properties These two operators are essentially adapted to the class of semi- strictly quasiconvex functions. Indeed this case, for each x ∈ < dom f \ arg min f, the sets Sf(x) and Sf(x) have the same closure < and Nf (x) = Nf (x). – p.10/48 II Normal approach b- Adjusted sublevel sets and normal operator – p.11/48 Definition Adjusted sublevel set For any x ∈ dom f, we define a < Sf (x) = Sf(x) ∩ B(Sf(x), ρx) < < where ρx = dist(x, Sf(x)), if Sf(x) =6 ∅. – p.12/48 Definition Adjusted sublevel set For any x ∈ dom f, we define a < Sf (x) = Sf(x) ∩ B(Sf(x), ρx) < < where ρx = dist(x, Sf(x)), if Sf(x) =6 ∅. a > Sf (x) coincides with Sf(x) if cl(Sf(x)) = Sf(x) e.g. f is semistrictly quasiconvex – p.12/48 Definition Adjusted sublevel set For any x ∈ dom f, we define a < Sf (x) = Sf(x) ∩ B(Sf(x), ρx) < < where ρx = dist(x, Sf(x)), if Sf(x) =6 ∅. a > Sf (x) coincides with Sf(x) if cl(Sf(x)) = Sf(x) Proposition 1 Let f : X → IR ∪ {+∞} be any function, with domain dom f. Then a f is quasiconvex ⇐⇒ Sf (x) is convex , ∀ x ∈ dom f. – p.12/48 Proof a Let us suppose that Sf (u) is convex for every u ∈ dom f. We will show that for any x ∈ dom f, Sf(x) is convex. a If x ∈ arg min f then Sf(x) = Sf (x) is convex by assumption. Assume now that x∈ / arg min f and take y, z ∈ Sf(x). If both y and z belong to B S< ,ρ , then y, z ∈ Sa (x) thus [y, z] ⊆ Sa (x) ⊆ S . “ f(x) x” f f f(x) If both y and z do not belong to B S< ,ρ , then “ f(x) x” < < < f(x)= f(y)= f(z), Sf(z) = Sf(y) = Sf(x) and ρ ,ρ are positive. If, say, ρ ≥ ρ then y, z ∈ B S< ,ρ thus y z y z “ f(y) y” a a y, z ∈ Sf (y) and [y, z] ⊆ Sf (y) ⊆ Sf(y) = Sf(x). – p.13/48 Proof < < Finally, suppose that only one of y, z, say z, belongs to B(Sf(x),ρx) while y∈ / B(Sf(x),ρx). Then < < f(x)= f(y), Sf(y) = Sf(x) and ρy >ρx so we have z ∈ B S< ,ρ ⊆ B S< ,ρ and we deduce as before that “ f(x) x” “ f(y) y” a [y, z] ⊆ Sf (y) ⊆ Sf(y) = Sf(x). The other implication is straightforward. – p.14/48 Adjusted normal operator Adjusted sublevel set: For any x ∈ dom f, we define a < Sf (x) = Sf(x) ∩ B(Sf(x), ρx) < < where ρx = dist(x, Sf(x)), if Sf(x) =6 ∅. Ajusted normal operator: a ∗ ∗ ∗ a Nf (x) = {x ∈ X : hx ,y − xi≤ 0, ∀ y ∈ Sf (x)} – p.15/48 Example x © – p.16/48 Example x © < B(Sf(x), ρx) a < Sf (x)= Sf (x) ∩ B(Sf(x), ρx) – p.16/48 Example a < Sf (x)= Sf (x) ∩ B(Sf(x), ρx) a ∗ ∗ ∗ a Nf (x)= {x ∈ X : hx , y − xi≤ 0, ∀ y ∈ Sf (x)} – p.16/48 Subdifferential vs normal operator One can have a a Nf (x) ⊆/ cone(∂f(x)) or cone(∂f(x)) ⊆/ Nf (x) Proposition 2 f is quasiconvex and x ∈ dom f If there exists δ > 0 such that 0 6∈ ∂Lf(B(x,δ)) then L ∞ a cone(∂ f(x)) ∪ ∂ f(x) ⊂ Nf (x). – p.17/48 a Basic properties of Nf Nonemptyness: Proposition 3 Let f : X → IR ∪ {+∞} be lsc. Assume that rad. continuous on dom f or dom f is convex and intSλ =6 ∅, ∀ λ > infX f. Then a If f is quasiconvex , Nf (x) \ {0}= 6 ∅, ∀ x ∈ dom f \ arg min f. f quasiconvex a ⇐⇒ dom Nf \ {0} dense in dom f \ arg min f. Quasimonotonicity: a The normal operator Nf is always quasimonotone – p.18/48 Upper sign-continuity ∗ • T : X → 2X is said to be upper sign-continuous on K iff for any x,y ∈ K, one have : ∀ t ∈ ]0, 1[, inf hx∗,y − xi≥ 0 ∗ x ∈T (xt) =⇒ sup hx∗,y − xi≥ 0 x∗∈T (x) where xt = (1 − t)x + ty. – p.19/48 Upper sign-continuity ∗ • T : X → 2X is said to be upper sign-continuous on K iff for any x,y ∈ K, one have : ∀ t ∈ ]0, 1[, inf hx∗,y − xi≥ 0 ∗ x ∈T (xt) =⇒ sup hx∗,y − xi≥ 0 x∗∈T (x) where xt = (1 − t)x + ty. upper semi-continuous ⇓ upper hemicontinuous ⇓ upper sign-continuous – p.19/48 locally upper sign continuity ∗ Definition 5 Let T : K → 2X be a set-valued map. T est called locally upper sign-continuous on K if, for any x ∈ K there exist a neigh. Vx of x and a upper sign-continuous set-valued X∗ ∗ map Φx(·) : Vx → 2 with nonempty convex w -compact values such that Φx(y) ⊆ T (y) \ {0}, ∀ y ∈ Vx – p.20/48 locally upper sign continuity ∗ Definition 6 Let T : K → 2X be a set-valued map. T est called locally upper sign-continuous on K if, for any x ∈ K there exist a neigh. Vx of x and a upper sign-continuous set-valued X∗ ∗ map Φx(·) : Vx → 2 with nonempty convex w -compact values such that Φx(y) ⊆ T (y) \ {0}, ∀ y ∈ Vx Continuity of normal operator Proposition 7 Let f be lsc quasiconvex function such that int(Sλ) =6 ∅, ∀ λ > inf f. Then Nf is locally upper sign-continuous on dom f \ arg min f. – p.20/48 Proposition 8 If f is quasiconvex such that int(Sλ) =6 ∅, ∀ λ> inf f and f is lsc at x ∈ dom f \ arg min f, a ∗ Then Nf is norm-to-w cone-usc at x. ∗ A multivalued map with conical valued T : X → 2X is said to be cone-usc at x ∈ dom T if there exists a neighbourhood U of x and a base C (u) of T (u), u ∈ U, such that u → C (u) is usc at x. – p.21/48 a Integration of Nf Let f : X → IR ∪ {+∞} quasiconvex Question: Is it possible to characterize the functions a a g : X → IR ∪ {+∞} quasiconvex such that Nf = Ng ? – p.22/48 a Integration of Nf Let f : X → IR ∪ {+∞} quasiconvex Question: Is it possible to characterize the functions a a g : X → IR ∪ {+∞} quasiconvex such that Nf = Ng ? A first answer: Let C = {g : X → IR ∪ {+∞} cont.
Load more

Recommended publications
  • Robustly Quasiconvex Function
    Robustly quasiconvex function Bui Thi Hoa Centre for Informatics and Applied Optimization, Federation University Australia May 21, 2018 Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 1 / 15 Convex functions 1 All lower level sets are convex. 2 Each local minimum is a global minimum. 3 Each stationary point is a global minimizer. Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 2 / 15 Definition f is called explicitly quasiconvex if it is quasiconvex and for all λ 2 (0; 1) f (λx + (1 − λ)y) < maxff (x); f (y)g; with f (x) 6= f (y): Example 1 f1 : R ! R; f1(x) = 0; x 6= 0; f1(0) = 1. 2 f2 : R ! R; f2(x) = 1; x 6= 0; f2(0) = 0. 3 Convex functions are quasiconvex, and explicitly quasiconvex. 4 f (x) = x3 are quasiconvex, and explicitly quasiconvex, but not convex. Generalised Convexity Definition A function f : X ! R, with a convex domf , is called quasiconvex if for all x; y 2 domf , and λ 2 [0; 1] we have f (λx + (1 − λ)y) ≤ maxff (x); f (y)g: Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 3 / 15 Example 1 f1 : R ! R; f1(x) = 0; x 6= 0; f1(0) = 1. 2 f2 : R ! R; f2(x) = 1; x 6= 0; f2(0) = 0. 3 Convex functions are quasiconvex, and explicitly quasiconvex. 4 f (x) = x3 are quasiconvex, and explicitly quasiconvex, but not convex. Generalised Convexity Definition A function f : X ! R, with a convex domf , is called quasiconvex if for all x; y 2 domf , and λ 2 [0; 1] we have f (λx + (1 − λ)y) ≤ maxff (x); f (y)g: Definition f is called explicitly quasiconvex if it is quasiconvex and
    [Show full text]
  • Local Maximum Points of Explicitly Quasiconvex Functions
    Local maximum points of explicitly quasiconvex functions Item Type Article Authors Bagdasar, Ovidiu; Popovici, Nicolae Citation Bagdasar, O. & Popovici, (2015) 'Local maximum points of explicitly quasiconvex functions' Optimization Letters, 9: 769. doi:10.1007/s11590-014-0781-3 DOI 10.1007/s11590-014-0781-3 Publisher Springer Journal Optimization Letters Rights Archived with thanks to Optimization Letters Download date 25/09/2021 20:26:30 Link to Item http://hdl.handle.net/10545/620886 Optimization Letters manuscript No. (will be inserted by the editor) Local maximum points of explicitly quasiconvex functions Ovidiu Bagdasar · Nicolae Popovici Received: date / Accepted: date Abstract This work concerns (generalized) convex real-valued functions defined on a nonempty convex subset of a real topological linear space. Its aim is twofold. The first concerns explicitly quasiconvex functions. As a counterpart of some known results, it is shown that any local maximum point of such a function is actually a global minimum point whenever it belongs to the intrinsic core of the function's domain. Secondly, we establish a new characterization of strictly convex normed spaces by applying this property for a particular class of convex functions. Keywords Local maximum point · Relative algebraic interior · Convex function · Explicitly quasiconvex function · Strictly convex space · Least squares problem 1 Introduction Optimization problems involving explicitly quasiconvex objective functions, i.e., real-valued functions which are both quasiconvex and semistrictly quasiconvex, have been intensively studied in the literature, beginning with the pioneering works by Martos [6] and Karamardian [5]. These functions are of special interest since they preserve several fundamental properties of convex functions.
    [Show full text]
  • A Class of Functions That Are Quasiconvex but Not Polyconvex
    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 12-2003 A Class of Functions That Are Quasiconvex But Not Polyconvex Catherine S. Remus University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Mathematics Commons Recommended Citation Remus, Catherine S., "A Class of Functions That Are Quasiconvex But Not Polyconvex. " Master's Thesis, University of Tennessee, 2003. https://trace.tennessee.edu/utk_gradthes/2219 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Catherine S. Remus entitled "A Class of Functions That Are Quasiconvex But Not Polyconvex." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics. Henry C. Simpson, Major Professor We have read this thesis and recommend its acceptance: Charles Collins, G. Samuel Jordan Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a thesis written by Catherine S. Remus entitled “A Class of Functions That Are Quasiconvex But Not Polyconvex”.
    [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]
  • 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]
  • NOTE on the CONTINUITY of M-CONVEX and L-CONVEX FUNCTIONS in CONTINUOUS VARIABLES 1. Introduction Two Kinds of Convexity Concept
    Journal of the Operations Research Society of Japan 2008, Vol. 51, No. 4, 265-273 NOTE ON THE CONTINUITY OF M-CONVEX AND L-CONVEX FUNCTIONS IN CONTINUOUS VARIABLES Kazuo Murota Akiyoshi Shioura University of Tokyo Tohoku University (Received March 27, 2008) Abstract M-convex and L-convex functions in continuous variables constitute subclasses of convex func- tions with nice combinatorial properties. In this note we give proofs of the fundamental facts that closed proper M-convex and L-convex functions are continuous on their effective domains. Keywords: Combinatorial optimization, convex function, continuity, submodular func- tion, matroid. 1. Introduction Two kinds of convexity concepts, called M-convexity and L-convexity, play primary roles in the theory of discrete convex analysis [6]. They are originally introduced for functions in integer variables by Murota [4, 5], and then for functions in continuous variables by Murota{ Shioura [8, 10]. M-convex and L-convex functions in continuous variables constitute subclasses of convex functions with additional combinatorial properties such as submodularity and diagonal dom- inance (see, e.g., [6{11]). Fundamental properties of M-convex and L-convex functions are investigated in [9], such as equivalent axioms, subgradients, directional derivatives, etc. Con- jugacy relationship between M-convex and L-convex functions under the Legendre-Fenchel transformation is shown in [10]. Subclasses of M-convex and L-convex functions are investi- gated in [8] (polyhedral M-convex and L-convex functions) and in [11] (quadratic M-convex and L-convex functions). As variants of M-convex and L-convex functions, the concepts of M\-convex and L\-convex functions are also introduced by Murota{Shioura [8, 10], where \M\" and \L\" should be read \M-natural" and \L-natural," respectively.
    [Show full text]
  • Characterisations of Quasiconvex Functions
    BULL. AUSTRAL. MATH. SOC. 26B25, 49J52 VOL. 48 (1993) [393-406] CHARACTERISATIONS OF QUASICONVEX FUNCTIONS DINH THE LUC In this paper we introduce the concept of quasimonotone maps and prove that a lower semicontinuous function on an infinite dimensional space is quasiconvex if and only if its generalised subdifferential or its directional derivative is quasimonotone. 1. INTRODUCTION As far as we know quasiconvex functions were first mentioned by de Finetti in his work "Sulle Straficazioni Convesse" in 1949, [6]. Since then much effort has been focused on the study of this class of functions for they have much in common with convex functions and they are used in several areas of science including mathematics, operations research, economics et cetera (see [15]). One of the most important properties of convex functions is that their level sets are convex. This property is also a fundamental geometric characterisation of quasiconvex functions which sometimes is treated as their definition. However, the most attractive characterisations of quasiconvex functions are those which involve gradients (a detailed account of the current state of research on the topic can be found in [l]). As to generalised derivatives of quasiconvex functions, very few results exist (see [1,7]). In [7], a study of quasiconvex functions is presented via Clarke's subdifferential, but the authors restrict themselves to the case of Lipschitz functions on a finite dimensional space only. The aim of our paper is to characterise lower semicontinuous quasiconvex functions in terms of generalised (Clarke-Rockafellar) subdifferentials and directional derivatives in infinite dimensional spaces. Namely, we shall first introduce a new concept of quasi- monotone maps; and then we prove the equivalence between the quasiconvexity of a function and the quasimonotonicity of its generalised subdifferential and its directional derivative.
    [Show full text]
  • Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach1
    JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 97, No. I, pp. 29-45, APRIL 1998 Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach1 D. AUSSEL2 Communicated by M. Avriel Abstract. In this paper, we are mainly concerned with the characteriza- tion of quasiconvex or pseudoconvex nondifferentiable functions and the relationship between those two concepts. In particular, we charac- terize the quasiconvexity and pseudoconvexity of a function by mixed properties combining properties of the function and properties of its subdifferential. We also prove that a lower semicontinuous and radially continuous function is pseudoconvex if it is quasiconvex and satisfies the following optimality condition: 0sdf(x) =f has a global minimum at x. The results are proved using the abstract subdifferential introduced in Ref. 1, a concept which allows one to recover almost all the subdiffer- entials used in nonsmooth analysis. Key Words. Nonsmooth analysis, abstract subdifferential, quasicon- vexity, pseudoconvexity, mixed property. 1. Preliminaries Throughout this paper, we use the concept of abstract subdifferential introduced in Aussel et al. (Ref. 1). This definition allows one to recover almost all classical notions of nonsmooth analysis. The aim of such an approach is to show that a lot of results concerning the subdifferential prop- erties of quasiconvex and pseudoconvex functions can be proved in a unified way, and hence for a large class of subdifferentials. We are interested here in two aspects of convex analysis: the charac- terization of quasiconvex and pseudoconvex lower semicontinuous functions and the relationship between these two concepts. 1The author is indebted to Prof. M. Lassonde for many helpful discussions leading to a signifi- cant improvement in the presentation of the results.
    [Show full text]
  • Chapter 5 Convex Optimization in Function Space 5.1 Foundations of Convex Analysis
    Chapter 5 Convex Optimization in Function Space 5.1 Foundations of Convex Analysis Let V be a vector space over lR and k ¢ k : V ! lR be a norm on V . We recall that (V; k ¢ k) is called a Banach space, if it is complete, i.e., if any Cauchy sequence fvkglN of elements vk 2 V; k 2 lN; converges to an element v 2 V (kvk ¡ vk ! 0 as k ! 1). Examples: Let ­ be a domain in lRd; d 2 lN. Then, the space C(­) of continuous functions on ­ is a Banach space with the norm kukC(­) := sup ju(x)j : x2­ The spaces Lp(­); 1 · p < 1; of (in the Lebesgue sense) p-integrable functions are Banach spaces with the norms Z ³ ´1=p p kukLp(­) := ju(x)j dx : ­ The space L1(­) of essentially bounded functions on ­ is a Banach space with the norm kukL1(­) := ess sup ju(x)j : x2­ The (topologically and algebraically) dual space V ¤ is the space of all bounded linear functionals ¹ : V ! lR. Given ¹ 2 V ¤, for ¹(v) we often write h¹; vi with h¢; ¢i denoting the dual product between V ¤ and V . We note that V ¤ is a Banach space equipped with the norm j h¹; vi j k¹k := sup : v2V nf0g kvk Examples: The dual of C(­) is the space M(­) of Radon measures ¹ with Z h¹; vi := v d¹ ; v 2 C(­) : ­ The dual of L1(­) is the space L1(­). The dual of Lp(­); 1 < p < 1; is the space Lq(­) with q being conjugate to p, i.e., 1=p + 1=q = 1.
    [Show full text]
  • Set Optimization with Respect to Variable Domination Structures
    Set optimization with respect to variable domination structures Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) der Naturwissenschaftlichen Fakult¨atII Chemie, Physik und Mathematik der Martin-Luther-Universit¨at Halle-Wittenberg vorgelegt von Frau Le Thanh Tam geb. am 05.05.1985 in Phu Tho Gutachter: Frau Prof. Dr. Christiane Tammer (Martin-Luther-Universit¨atHalle-Wittenberg) Herr Prof. Dr. Akhtar Khan (Rochester Institute of Technology) Tag der Verteidigung: 02.10.2018 List of Figures K 2.1 Set relations t ; t 2 fl; u; pl; pu; cl; cug in the sense of Definition 2.2.5. 20 4.1 Illustration for Example 4.2.4......................... 45 4.2 Illustration for Theorems 4.3.1, 4.3.2 and 4.3.5............... 51 4.3 Illustration for Theorems 4.3.7 and 4.3.8.................. 53 5.1 Illustration for Example 5.1.7......................... 60 5.2 Illustration for Example 5.1.10........................ 61 5.3 Illustration for Example 5.1.11........................ 62 5.4 Illustration for Example 5.1.15........................ 65 5.5 A model of location optimization w.r.t. variable domination structures. 75 6.1 Illustration for Example 6.1.1......................... 87 6.2 Illustration for Example 6.1.2......................... 88 8.1 Dose response curves in lung cancer treatment............... 110 8.2 Discretization of patient into voxels and of beam into bixels, [32]..... 111 2 8.3 Visualization of two outcome sets A; Be ⊂ R of a uncertain bicriteria optimization problem with undesired elements............... 129 1 Contents 1 Introduction1 2 Preliminaries7 2.1 Binary Relations...............................7 2.1.1 Linear Spaces, Topological Vector Spaces and Normed Spaces..7 2.1.2 Order Relations on Linear Spaces.................
    [Show full text]
  • Chapter 2 Convex Functions
    Chapter 2 Convex functions 2.1 Basic properties and examples 2.1.1 Definition A function f : Rn → R is convex if dom f is a convex set and if for all x, y ∈ dom f,and θ with 0 ≤ θ ≤ 1, we have f(θx +(1− θ)y) ≤ θf(x)+(1− θ)f(y). (2.1) Geometrically, this inequality means that the line segment between (x, f(x)) and (y, f(y)) (i.e.,thechord from x to y) lies above the graph of f (figure 2.1). A function f is strictly convex if strict inequality holds in (2.1) whenever x = y and 0 <θ<1. We say f is concave if −f is convex, and strictly concave if −f is strictly convex. For an affine function we always have equality in (2.1), so all affine (and therefore also linear) functions are both convex and concave. Conversely, any function that is convex and concaveisaffine. A function is convex if and only if it is convex when restricted to any line that intersects its domain. In other words f is convex if and only if for all x ∈ dom f and all v, the function (y, f(y)) (x, f(x)) Figure 2.1: Graph of a convex function. The chord (i.e., line segment) between any two points on the graph lies above the graph. 43 44 CHAPTER 2. CONVEX FUNCTIONS h(t)=f(x + tv) is convex (on its domain, {t | x + tv ∈ dom f}). This property is very useful, since it allows us to check whether a function is convex by restrictingit to a line.
    [Show full text]
  • Arxiv:2104.13510V2 [Math.FA]
    Fenchel-Rockafellar Theorem in Infinite Dimensions via Generalized Relative Interiors D. V. Cuong 1,2, B. S. Mordukhovich3, N. M. Nam4, G. Sandine5 Abstract. In this paper we provide further studies of the Fenchel duality theory in the general framework of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong duality under some qualification conditions via generalized relative interiors imposed on the epigraphs and the domains of the functions involved. Our results directly generalize the classical Fenchel-Rockafellar theorem on strong duality from finite dimensions to LCTV spaces. Key words. Relative interior, quasi-relative interior, intrinsic relative interior, quasi-regularity, Fenchel duality. AMS subject classifications. 49J52, 49J53, 90C31 1 Introduction Duality theory has a central role in optimization theory and its applications. From a primal optimization problem with the objective function defined in a primal space, a dual problem in the dual space is formulated with the hope that the new problem is easier to solve, while having a close relationship with the primal problem. Its important role in optimization makes the duality theory attractive for extensive research over the past few decades; see, e.g., [1, 2, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19, 20, 25, 26, 28] and the references therein. Given a proper convex function f and a proper concave function g defined on Rn, the classical Fenchel-Rockafellar theorem asserts that inf f(x) g(x) x Rn = sup g (x∗) f ∗(x∗) x∗ Rn , (1.1) { − | ∈ } { ∗ − | ∈ } under the assumption that ri (dom (f)) ri (dom (g)) = ; see [27, Theorem 31.1].
    [Show full text]