DOCSLIB.ORG

A Class of Functions That Are Quasiconvex but Not Polyconvex

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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”. 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__________ Acceptance for the Council: Anne Mayhew___________ Vice Provost and Dean of Graduate Studies (Original signatures are on file with official student records.) A Class of Functions That Are Quasiconvex But Not Polyconvex A Thesis Presented for the Masters of Science Degree The University of Tennessee, Knoxville Catherine S. Remus December 2003 Acknowledgements I would like to thank everyone who helped me earn the Master of Science degree and master the discipline of Mathematics. I wish to thank Dr. Simpson for sharing his vast knowledge of Convex Analysis with me, but mostly for opening my eyes to the beauty of theoretical Mathematics and for drawing out my analytic abilities. I want to thank Dr. Jordan for pulling together all the mathematical concepts and tools I had learned into one integrated body of knowledge and for introducing me to the Calculus of Variations that led me to my thesis topic. I would also like to thank Dr. Collins for serving on my committee. I wish to thank my mother and son for their unflagging support and love during this journey of discovery I have taken us on. Even when I doubted, they never did. Finally, I thank God for all His Grace in guiding me and wish always to follow where He leads. ii Abstract In 1991 V. Sverak [11] gave an example of a function that was invariant and quasiconvex but not polyconvex. We have generalized this example to a wide class of functions that meet certain ellipticity and growth conditions. Quasiconvexity is one necessary and sufficient condition for the existence of solutions to the minimization problem in elliptic P.D.E. theory. Invariance is frequently a requirement of the stored energy function in Calculus of Variation approaches to elasticity problems. iii Table of Contents Chapter Page I Introduction ………………………………………………………………….. 1 II Background ……………………………………………………………...…… 3 III Theorem 1 ………………………………………………………………...…. 12 IV Theorem 2 ………………………………………………………………….... 16 V Theorem 3 ………………………………………………………………….... 17 VI Examples ……………………………………………………………….……. 21 List of References ………………………………………………………….. 27 Vita ……………………………………………………………………………. 30 iv Chapter I Introduction In this thesis a class of functions which are continuous, subject to certain growth and ellipticity conditions, and quasiconvex are shown not to be polyconvex. The study of convex analysis concerns itself with four types of convexity: convex functions, polyconvex functions, quasiconvex functions, and rank-one convex functions. The easiest way to visualize the relationships between these types of convexity is the analogy of nested sets where convex functions is the smallest, most stringent, set and rank-one convex functions is the largest, least stringent, set. The “proper subset” relationship of these convexity conditions has been a source of study for some years. In applications, quasiconvexity is very difficult to determine whereas polyconvexity is easier to show and the two concepts are near each other. V. Sverak [11] gave a new example of a function that is quasiconvex but not polyconvex in 1991. Other examples have been given in the past, but his new example was suggestive of an entire class of such functions. We construct a wide class of functions that are quasiconvex but not polyconvex. Sverak’s example is a special case of our examples. Quasiconvexity is a necessary and sufficient condition (along with others) to ensure the existence of solutions to the minimization problem in elliptic P.D.E. theory. Techniques of variational calculus have long been utilized in elasticity problems and the stored energy function, our f , is often required to be objective, i.e., frame indifferent or Galilean invariant, and isotropic, i.e., rotationally invariant (see Dacorogna [5]). 1 In Chapter II we give background information such as definitions, notation, and a lemma needed in the proof of Theorem 1. Chapter III states and proves a needed preliminary theorem, Theorem 1, guaranteeing specific growth and quasiconvexity conditions on our class of functions. In Chapter IV we have the primary emphasis of this thesis, Theorem 2. This theorem deals with the nonpolyconvexity of a class of quasiconvex functions subject to subquadratic growth and ellipticity conditions. Chapter V presents Theorem 3 that addresses the invariance of functions in our class of quasiconvex functions. Finally, in Chapter VI we give four examples of functions in this class of functions we have constructed. Three of the four examples meet the criteria of all three of our theorems, but Example 2 is convex and therefore polyconvex and does not meet the criteria for Theorem 2. This alleged failure highlights the difficulty in constructing nontrivial examples of functions in this class satisfying all three theorems. 2 Chapter II Background In this thesis, + is the set of positive real numbers and M n×n is the space of all n× n real matrices, n ≥1, and ⋅ is any norm on this space. The 4-tensor, C , is a linear n transformation M n×n → M n×n and is of the general form C [ X ] = C X , ij ijkl kl k,l=1 n×n n ∞ n ¡ ¡ ∈ ¡ ∈ Ω ⊂ φ ∈ (Ω) Cijkl , for all X M . For , C with values in , we denote £ ¢ ∂φ ∂φ ¤ ¥ 1 ( x) ¨ 1 ( x) ¥ ¤ ∂ ∂ x1 xn n ¤ ¥ © ∂φ k ¥ ∇φ ( x) = ¤ , so that C ∇φ ( x) = C (x) . ij ijkl ∂ ¥ ¤ x ∂φ ∂φ k,l=1 l ¤ ¥ n ( ) ¨ n ( ) § ¦ x x ∂ ∂ x1 xn n×n For this work we will be considering functions f : M → ¡ having various convexity properties. These are seen in physical theories such as elasticity, phase transition and many other applications that are considered the minimization of energy functionals, f (∇φ ( x))dx . The function f represents a potential energy function and Ω φ is the deformation variable. There exist many theorems discussing this type of function, f , and all require some kind of convexity and associated conditions, e.g., growth and coercivity estimates (see Dacorogna [5]). 3 Definitions and Notation n > ∀ ∈ n { } C is strongly elliptic if and only if Cijkl aibjakbl 0 a,b \ 0 i, j,k,l=1 (Note this is a strengthened Legendre-Hadamard condition; the Legendre-Hadamard condition has ≥ in place of >.) (see Gelfand-Fomin [7], Wan [12], Morrey [9]). n Let K ⊂ ¡ . K is a convex set if, for any two points x, y ∈ K and t ∈[0,1] the point z = tx + (1− t) y is also in K . ⊂ n = Let K ¡ . The convex hull of K , co(K ) : the smallest convex set that contains K = {tx + (1− t) y | x, y ∈ K, t ∈[0,1]} . A function g : n → is convex if and only if, for all x, y ∈ n and for all t ∈[0,1], g (tx + (1− t) y) ≤ tg ( x) + (1− t) g ( y) . For the following several definitions, let f : M n×n → and let Ω be the open unit ball in n . The function f is rank-one convex if and only if, for every A∈ M n×n and for all ∈ n ¡ + ⊗ ∈ ⊗ a,b the mapping t f ( A ta b) , t ¢ is convex. Here, a b is the element of n×n T M with components (a ⊗b) = a b , i.e., a ⊗b = a b ; this is the general rank one ij i j n× n matrix (provided either a or b is nonzero). The function f is quasiconvex if and only if, for every A∈ M n×n and for every φ ∈ ∞ (Ω) ( ) ≤ + ∇φ ( ) C0 , f A dx f ( A x )dx is true. Ω Ω 4 n×n The function f is polyconvex if and only if, for every A∈ M , there exists a τ n → = ¢ convex function g mapping ¢ , such that f ( A) g (T ( A)) where × τ n n → n = ¢ ( ) ( ) T : M and T is defined by T A : A, adj2 A, , adjn A . Here, adjs A is the n × ∈ n×n ≤ ≤ τ = σ adjugate matrix of all s s minors of the matrix A M , 2 s n , and n s s=1 ¢ £ £ ¢ £ ¢ 2 ¤ ¥ ¥ ¤ ¥ ¤ n n ¥ σ = = ¤ n! ¦ § ¦ § § where ¦ .
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]
  • 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 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]
  • Glimpses Upon Quasiconvex Analysis Jean-Paul Penot
    Glimpses upon quasiconvex analysis Jean-Paul Penot To cite this version: Jean-Paul Penot. Glimpses upon quasiconvex analysis. 2007. hal-00175200 HAL Id: hal-00175200 https://hal.archives-ouvertes.fr/hal-00175200 Preprint submitted on 27 Sep 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ESAIM: PROCEEDINGS, Vol. ?, 2007, 1-10 Editors: Will be set by the publisher DOI: (will be inserted later) GLIMPSES UPON QUASICONVEX ANALYSIS Jean-Paul Penot Abstract. We review various sorts of generalized convexity and we raise some questions about them. We stress the importance of some special subclasses of quasiconvex functions. Dedicated to Marc Att´eia R´esum´e. Nous passons en revue quelques notions de convexit´eg´en´eralis´ee.Nous tentons de les relier et nous soulevons quelques questions. Nous soulignons l’importance de quelques classes particuli`eres de fonctions quasiconvexes. 1. Introduction Empires usually are well structured entities, with unified, strong rules (for instance, the length of axles of carts in the Chinese Empire and the Roman Empire, a crucial rule when building a road network). On the contrary, associated kingdoms may have diverging rules and uses.
    [Show full text]
  • CONVEX SETS and CONCAVE FUNCTIONS W. Erwin Diewert
    1 ECONOMICS 594: LECTURE NOTES CHAPTER 2: CONVEX SETS AND CONCAVE FUNCTIONS W. Erwin Diewert January 31, 2008. 1. Introduction Many economic problems have the following structure: (i) a linear function is minimized subject to a nonlinear constraint; (ii) a linear function is maximized subject to a nonlinear constraint or (iii) a nonlinear function is maximized subject to a linear constraint. Examples of these problems are: (i) the producer’s cost minimization problem (or the consumer’s expenditure minimization problem); (ii) the producer’s profit maximization problem and (iii) the consumer’s utility maximization problem. These three constrained optimization problems play a key role in economic theory. In each of the above 3 problems, linear functions appear in either the objective function (the function being maximized or minimized) or the constraint function. If we are N T maximizing or minimizing the linear function of x, say n=1 pnxn p x, where p 1 (p1,…,pN) is a vector of prices and x (x1,…,xN) is a vector of decision variables, then after the optimization problem is solved, the optimized objective function can be regarded as a function of the price vector, say G(p), and perhaps other variables that appear in the constraint function. Let F(x) be the nonlinear function which appears in the constraint in problems (i) and (ii). Then under certain conditions, the optimized objective function G(p) can be used to reconstruct the nonlinear constraint function F(x). This correspondence between F(x) and G(p) is known as duality theory. In the following chapter, we will see how the use of dual functions can greatly simplify economic modeling.
    [Show full text]
  • Generalized Convexity and Optimization
    Lecture Notes in Economics and Mathematical Systems 616 Generalized Convexity and Optimization Theory and Applications Bearbeitet von Alberto Cambini, Laura Martein 1. Auflage 2008. Taschenbuch. xiv, 248 S. Paperback ISBN 978 3 540 70875 9 Format (B x L): 15,5 x 23,5 cm Gewicht: 405 g Wirtschaft > Betriebswirtschaft: Theorie & Allgemeines > Wirtschaftsmathematik und - statistik Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. 2 Non-Differentiable Generalized Convex Functions 2.1 Introduction In several economic models convexity appears to be a restrictive condition. For instance, classical assumptions in Economics include the convexity of the production set in producer theory and the convexity of the upper level sets of the utility function in consumer theory. On the other hand, in Optimization it is important to know when a local minimum (maximum) is also global. Such a useful property is not exclusive to convexity (concavity). All this has led to the introduction of new classes of functions which have con- vex lower/upper level sets and which verify the local-global property, starting with the pioneer work of Arrow–Enthoven [7]. In this chapter we shall introduce the class of quasiconvex, strictly quasicon- vex, and semistrictly quasiconvex functions and the inclusion relationships between them are studied. For functions in one variable, a complete charac- terization of the new classes is established.
    [Show full text]
  • Risk- and Ambiguity-Averse Portfolio Optimization with Quasiconcave Utility Functionals
    Finance Stoch (2017) 21:397–425 DOI 10.1007/s00780-016-0318-y Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals Sigrid Källblad1 Received: 25 November 2014 / Accepted: 15 February 2016 / Published online: 15 March 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract Motivated by recent axiomatic developments, we study the risk- and ambi- guity-averse investment problem where trading takes place in continuous time over a fixed finite horizon and terminal payoffs are evaluated according to criteria defined in terms of quasiconcave utility functionals. We extend to the present setting certain ex- istence and duality results established for so-called variational preferences by Schied (Finance Stoch. 11:107–129, 2007). The results are proved by building on existing re- sults for the classical utility maximization problem, combined with a careful analysis of the involved quasiconvex and semicontinuous functions. Keywords Model uncertainty · Ambiguity aversion · Quasiconvex risk measures · Optimal investment · Robust portfolio selection · Duality theory Mathematics Subject Classification (2010) 49N15 · 62C20 · 91B06 · 91B16 · 91G10 · 91G80 JEL Classification G11 · C61 · D81 1 Introduction The optimal investment problem of choosing the best way to allocate investors’ cap- ital is often formulated as the problem of maximizing, over admissible investment strategies, the expected utility of terminal wealth. The formulation relies on the ax- iomatic foundation developed by von Neumann and Morgenstern [77] and Savage B S. Källblad [email protected] 1 Institut für Stochastik und Wirtschaftsmathematik, Technische Universität Wien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria 398 S. Källblad [70]. In continuous-time optimal portfolio selection, the study dates back to the sem- inal contributions of Merton [60, 61].
    [Show 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)}.
    [Show full text]
  • Quasiconvex Programming
    Combinatorial and Computational Geometry MSRI Publications Volume 52, 2005 Quasiconvex Programming DAVID EPPSTEIN Abstract. We define quasiconvex programming, a form of generalized lin- ear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics. 1. Introduction Quasiconvex programming is a form of geometric optimization, introduced in [Amenta et al. 1999] in the context of mesh improvement techniques and since applied to other problems in meshing, scientific computation, information visual- ization, automated algorithm analysis, and robust statistics [Bern and Eppstein 2001; 2003; Chan 2004; Eppstein 2004]. If a problem can be formulated as a quasiconvex program of bounded dimension, it can be solved algorithmically in a linear number of constant-complexity primitive operations by generalized linear programming techniques, or numerically by generalized gradient descent techniques. In this paper we survey quasiconvex programming algorithms and applications. 1.1. Quasiconvex functions. Let Y be a totally ordered set, for instance the real numbers R or integers Z ordered numerically. For any function f : X 7→ Y , and any value λ ∈ Y , we define the lower level set f ≤λ = {x ∈ X | f(x) ≤ λ} . A function q : X 7→ Y , where X is a convex subset of Rd, is called quasiconvex [Dharmadhikari and Joag-Dev 1988] when its lower level sets are all convex. A one-dimensional quasiconvex function is more commonly called unimodal, and This research was supported in part by NSF grant CCR-9912338.
    [Show full text]
  • The 7Thinternational Symposium on Generalized Convexity/Monotonicity
    The 7th International Symposium on Generalized Convexity/Monotonicity August 27-31, 2002 Hanoi, Vietnam Hanoi Institute of Mathematics The 7th International Symposium on Generalized Convexity/Monotonicity 3 Contents 1. Topics ....................................................................4 2. Organizers ................................................................5 3. Committees ...............................................................6 4. Sponsors ..................................................................8 5. List of Contributions .....................................................9 6. Abstracts.................................................................15 7. List of Participants ......................................................61 8. Index ....................................................................76 4 The 7th International Symposium on Generalized Convexity/Monotonicity Topics The symposium is aimed at bringing together researchers from all continents to report their latest results and to exchange new ideas in the field of generalized convexity and generalized monotonicity and their applications in optimization, control, stochastic, economics, management science, finance, engineering and re- lated topics. The 7th International Symposium on Generalized Convexity/Monotonicity 5 Organizers • International Working Group on Generalized Convexity (WGGC) • Institute of Mathematics, Vietnam NCST Location The conference takes place at the Institute of Mathematics National Centre for Natural Sciences
    [Show full text]