DOCSLIB.ORG

Convex) Level Sets Integration

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Convex) Level Sets Integration Journal of Optimization Theory and Applications manuscript No. (will be inserted by the editor) (Convex) Level Sets Integration Jean-Pierre Crouzeix · Andrew Eberhard · Daniel Ralph Dedicated to Vladimir Demjanov Received: date / Accepted: date Abstract The paper addresses the problem of recovering a pseudoconvex function from the normal cones to its level sets that we call the convex level sets integration problem. An important application is the revealed preference problem. Our main result can be described as integrating a maximally cycli- cally pseudoconvex multivalued map that sends vectors or \bundles" of a Eu- clidean space to convex sets in that space. That is, we are seeking a pseudo convex (real) function such that the normal cone at each boundary point of each of its lower level sets contains the set value of the multivalued map at the same point. This raises the question of uniqueness of that function up to rescaling. Even after normalising the function long an orienting direction, we give a counterexample to its uniqueness. We are, however, able to show uniqueness under a condition motivated by the classical theory of ordinary differential equations. Keywords Convexity and Pseudoconvexity · Monotonicity and Pseudomono- tonicity · Maximality · Revealed Preferences. Mathematics Subject Classification (2000) 26B25 · 91B42 · 91B16 Jean-Pierre Crouzeix, Corresponding Author, LIMOS, Campus Scientifique des C´ezeaux,Universit´eBlaise Pascal, 63170 Aubi`ere,France, E-mail: [email protected] Andrew Eberhard School of Mathematical & Geospatial Sciences, RMIT University, Melbourne, VIC., Aus- tralia, E-mail: [email protected] Daniel Ralph University of Cambridge, Judge Business School, UK, E-mail: [email protected] 2 Jean-Pierre Crouzeix et al. 1 Introduction Given a (multivalued) map the convex integration problem consists of finding a convex function whose subdifferential, or gradient in the smooth case, coincides with the map. Under appropriate conditions, the solution is unique up to a constant. In this paper, one considers the much more difficult problem where we are seeking a pseudo convex function corresponding to a map that generates the normal cones to the level sets of the function. One question is the uniqueness of the solution up to a scalarization. In both cases, two situations are considered: the map is continuously dif- ferentiable and the map is multivalued. Despite the great difference of difficulties encountered, there are similari- ties in the treatment of the differentiable and multivalued cases in these two situations. 2 Statement of the problem Given a 2 C ⊂ IRn ; α 2 IR and F : C ! IRn , the classical integration problem consists of finding f : C ! IR differentiable on C such that rf(x) = F (x) for all x 2 C and f(a) = α. For convenience, we assume throughout all the paper that C is open and convex. When f turns out to be convex, hF (x); y − xi ≤ f(y) − f(x) 8 x; y 2 C: In the convex case, the generalization of the integration problem to multi- valued monotone maps F consists of finding f convex on C such that f(a) = α and hx∗; y − xi ≤ f(y) − f(x) 8 x; y 2 C; 8 x∗ 2 F (x): We speak of this problem as the convex integration problem. Another problem, significantly more difficult, consists, when F is single valued, of finding f ! IR differentiable such that rf(x) F (x) = 8 x 2 C: krf(x)k kF (x)k We speak of this problem as the differentiable level sets integration problem. In case where f turns out to be pseudoconvex, then hrf(x); y − xi ≤ 0 8 x 2 C; 8y 2 S(x); (Convex) Level Sets Integration 3 where S(x) is the level set S(x) := fy 2 C : f(y) ≤ f(x)g: We consider the generalisation of this problem to the case where F is multivalued. It consists of finding f pseudoconvex such that hx∗; y − xi ≤ 0 8 x 2 C; 8 y 2 S(x); 8 x∗ 2 F (x): We call this problem the convex level sets integration problem. If the theories of classical and convex integration have been well estab- lished over several decades, this is not yet the case for level sets integration. However, despite the different levels of difficulties inherent in these problems, the approaches of their respective treatments present strong similarities. It is but one of the purposes of this paper to show the similarities and the differ- ences between differentiable and non differentiable, convex and convex level sets integration. In this respect, Section 3 briefly discusses classical integration in order to prepare the treatment of (convex) level sets integration. Section 4, based on Crouzeix{Rapcs´ak[1], treats the differentiable level sets convex prob- lem for comparison with Section 6 where the nondifferentiable case is studied. Section 5 is a very short review of the generalized convex and monotone prop- erties used in Section 6. Finally, Section 7 connects the paper with the problem known in economics as the revealed preference problem. The inner product of two vectors x; y 2 IRn is denoted by hx; yi or x>y. The Euclidean norm of a vector x 2 IRn is denoted by kxk. The convex hull n n of a set A is denoted by conv (A). Given a multivalued map F : IR ⇒ IR its graph and its domain are the sets graph (F ) := f(x; x ∗) 2 IRn × IRn : x ∗ 2 F (x)g and dom (F ) := fx 2 IRn : F (x) 6= ;g respectively. For a set S ⊂ IRn × IRn , Proj (graph (F )) := fx 2 IRn : 9y 2 IRn ; (x; y) 2 Sg, e.g., dom (F ) = Proj (graph (F )). 3 Integration and Convex Integration Through all sections C ⊂ IRn is an open convex set. 3.1 The Differentiable Case Assume that F : C ! IR is continuous. Let a 2 C, and α 2 IR. The problem consists in finding a function f : C ! IR continuously differentiable on C such that rf(x) = F (x) for all x 2 C and f(a) = α. 4 Jean-Pierre Crouzeix et al. If such a f exists, then necessarily, Z 1 f(x) = α + hF (a + t(x − a)); x − aidt 8 x 2 C: (1) 0 Thus, if the problem admits one solution, the solution is unique. The existence of a solution is a little more difficult question, additional assumptions are re- quired. In order to prepare the treatment of differentiable level sets integration in Section 3, we briefly describe a sketch of the proof. The function f given by formula (1) is a solution of the problem if and only if for all d such that (a + d) 2 C 1 [f(a + d + h) − f(a) − hF (a + d); hi] ! 0 when h ! 0: (2) khk It is well known that this condition holds as soon as F is continuously differ- entiable on C and the matrix F 0(x) is symmetric for all x 2 C. Under this condition, f is convex if and only if F 0(x) is positive semidefinite for all x 2 C. Expression (1) provides a construction of the (convex) function f. To summarize: the existence of f such that F coincides with rf is ensured by the continuity of F 0 and the symmetry of the matrices F 0(x), the convexity of f by the positive semidefiniteness of the matrices F 0(x), the uniqueness of f up to a constant is ensured by the continuity of F . 3.2 The Nondifferentiable Case Next, we consider the generalization of the problem to nondifferentiable func- tions f. An appropriate generalization of differentiability for convex functions is the subdifferentiability in the sense of Fenchel. Recall that the subdifferential @f of a function f is defined by @f(x) := f x∗ : hx∗; y − xi ≤ f(y) − f(x) 8 yg: The subdifferential of a convex function is a cyclically monotone map, i.e., for ∗ ∗ ∗ all finite family f(xi; xi ); i = 0; ··· ; pg ⊂ graph (@f ) = f(x; x ): x 2 @f (x)g ∗ ∗ with (x0; x0) = (xp; xp) one has p−1 X ∗ hxi ; xi+1 − xii ≤ 0: i=0 If f : C ! IR is a differentiable convex function, then @f(x) = frf(x)g at all x 2 C. (Convex) Level Sets Integration 5 The convex integration problem consists in building a convex function f n from a cyclically monotone multivalued map F : C ⇒ IR such that f(a) = α and F (x) ⊂ @f(x) 8 x 2 dom (F ) := f y : 9 y∗ such that y∗ 2 F (x)g: (3) 3.2.1 Existence Let ; 6= S ⊂ graph (F ), (a; a∗) 2 S and α 2 IR. The following construction of n fS : IR ! IR [ f+1g is borrowed from Rockafellar [2]: 2 3 p p 2 N; xp+1 = x; X f (x) := α + sup 6 hx∗; x − x i : ∗ ∗ 7 : (4) S 4 i i+1 i (x0; x0) = (a; a ); 5 i=0 ∗ (xi; xi ) 2 S; i = 1; ··· ; p It straightforwardly follows that fS is a proper convex lower semicontinuous function because it is a supremum of affine functions. Also, fS ≤ fT whenever S ⊂ T ⊂ graph (F ). Next, due to the cyclic monotonicity of F , one has ∗ ∗ fS(a) = α and x 2 @fS(x) for all (x; x ) 2 S. Actually, fS is the smallest lower semicontinuous convex function for which these two conditions hold. Also, fS is finite on the interior of the convex hull of Proj (S). If card (S) is finite, then fS is a convex piecewise linear function. It can be explicitly computed through a finite number of steps.
Load more

Recommended publications
  • 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]
  • Lecture 5: Maxima and Minima of Convex Functions
    Advanced Mathematical Programming IE417 Lecture 5 Dr. Ted Ralphs IE417 Lecture 5 1 Reading for This Lecture ² Chapter 3, Sections 4-5 ² Chapter 4, Section 1 1 IE417 Lecture 5 2 Maxima and Minima of Convex Functions 2 IE417 Lecture 5 3 Minimizing a Convex Function Theorem 1. Let S be a nonempty convex set on Rn and let f : S ! R be ¤ convex on S. Suppose that x is a local optimal solution to minx2S f(x). ² Then x¤ is a global optimal solution. ² If either x¤ is a strict local optimum or f is strictly convex, then x¤ is the unique global optimal solution. 3 IE417 Lecture 5 4 Necessary and Sufficient Conditions Theorem 2. Let S be a nonempty convex set on Rn and let f : S ! R be convex on S. The point x¤ 2 S is an optimal solution to the problem T ¤ minx2S f(x) if and only if f has a subgradient » such that » (x ¡ x ) ¸ 0 8x 2 S. ² This implies that if S is open, then x¤ is an optimal solution if and only if there is a zero subgradient of f at x¤. 4 IE417 Lecture 5 5 Alternative Optima Theorem 3. Let S be a nonempty convex set on Rn and let f : S ! R be convex on S. If the point x¤ 2 S is an optimal solution to the problem minx2S f(x), then the set of alternative optima are characterized by the set S¤ = fx 2 S : rf(x¤)T (x ¡ x¤) · 0 and rf(x¤) = rf(x) Corollaries: 5 IE417 Lecture 5 6 Maximizing a Convex Function Theorem 4.
    [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]
  • 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]
  • 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 the Ekeland Variational Principle with Applications and Detours
    Lectures on The Ekeland Variational Principle with Applications and Detours By D. G. De Figueiredo Tata Institute of Fundamental Research, Bombay 1989 Author D. G. De Figueiredo Departmento de Mathematica Universidade de Brasilia 70.910 – Brasilia-DF BRAZIL c Tata Institute of Fundamental Research, 1989 ISBN 3-540- 51179-2-Springer-Verlag, Berlin, Heidelberg. New York. Tokyo ISBN 0-387- 51179-2-Springer-Verlag, New York. Heidelberg. Berlin. Tokyo No part of this book may be reproduced in any form by print, microfilm or any other means with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 Printed by INSDOC Regional Centre, Indian Institute of Science Campus, Bangalore 560012 and published by H. Goetze, Springer-Verlag, Heidelberg, West Germany PRINTED IN INDIA Preface Since its appearance in 1972 the variational principle of Ekeland has found many applications in different fields in Analysis. The best refer- ences for those are by Ekeland himself: his survey article [23] and his book with J.-P. Aubin [2]. Not all material presented here appears in those places. Some are scattered around and there lies my motivation in writing these notes. Since they are intended to students I included a lot of related material. Those are the detours. A chapter on Nemyt- skii mappings may sound strange. However I believe it is useful, since their properties so often used are seldom proved. We always say to the students: go and look in Krasnoselskii or Vainberg! I think some of the proofs presented here are more straightforward. There are two chapters on applications to PDE.
    [Show full text]
  • Optimization Over Nonnegative and Convex Polynomials with and Without Semidefinite Programming
    OPTIMIZATION OVER NONNEGATIVE AND CONVEX POLYNOMIALS WITH AND WITHOUT SEMIDEFINITE PROGRAMMING GEORGINA HALL ADISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF OPERATIONS RESEARCH AND FINANCIAL ENGINEERING ADVISER:PROFESSOR AMIR ALI AHMADI JUNE 2018 c Copyright by Georgina Hall, 2018. All rights reserved. Abstract The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, con- trol, machine learning, game theory, and combinatorics, among others. A number of break- through papers in the early 2000s showed that this problem, long thought to be out of reach, could be tackled by using sum of squares programming. This technique however has proved to be expensive for large-scale problems, as it involves solving large semidefinite programs (SDPs). In the first part of this thesis, we present two methods for approximately solving large- scale sum of squares programs that dispense altogether with semidefinite programming and only involve solving a sequence of linear or second order cone programs generated in an adaptive fashion. We then focus on the problem of finding tight lower bounds on polyno- mial optimization problems (POPs), a fundamental task in this area that is most commonly handled through the use of SDP-based sum of squares hierarchies (e.g., due to Lasserre and Parrilo). In contrast to previous approaches, we provide the first theoretical framework for constructing converging hierarchies of lower bounds on POPs whose computation simply requires the ability to multiply certain fixed polynomials together and to check nonnegativ- ity of the coefficients of their product.
    [Show full text]
  • Sum of Squares and Polynomial Convexity
    CONFIDENTIAL. Limited circulation. For review only. Sum of Squares and Polynomial Convexity Amir Ali Ahmadi and Pablo A. Parrilo Abstract— The notion of sos-convexity has recently been semidefiniteness of the Hessian matrix is replaced with proposed as a tractable sufficient condition for convexity of the existence of an appropriately defined sum of squares polynomials based on sum of squares decomposition. A multi- decomposition. As we will briefly review in this paper, by variate polynomial p(x) = p(x1; : : : ; xn) is said to be sos-convex if its Hessian H(x) can be factored as H(x) = M T (x) M (x) drawing some appealing connections between real algebra with a possibly nonsquare polynomial matrix M(x). It turns and numerical optimization, the latter problem can be re- out that one can reduce the problem of deciding sos-convexity duced to the feasibility of a semidefinite program. of a polynomial to the feasibility of a semidefinite program, Despite the relative recency of the concept of sos- which can be checked efficiently. Motivated by this computa- convexity, it has already appeared in a number of theo- tional tractability, it has been speculated whether every convex polynomial must necessarily be sos-convex. In this paper, we retical and practical settings. In [6], Helton and Nie use answer this question in the negative by presenting an explicit sos-convexity to give sufficient conditions for semidefinite example of a trivariate homogeneous polynomial of degree eight representability of semialgebraic sets. In [7], Lasserre uses that is convex but not sos-convex. sos-convexity to extend Jensen’s inequality in convex anal- ysis to linear functionals that are not necessarily probability I.
    [Show full text]
  • NMSA403 Optimization Theory – Exercises Contents
    NMSA403 Optimization Theory { Exercises Martin Branda Charles University, Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Version: December 17, 2017 Contents 1 Introduction and motivation 2 1.1 Operations Research/Management Science and Mathematical Program- ming . .2 1.2 Marketing { Optimization of advertising campaigns . .2 1.3 Logistic { Vehicle routing problems . .2 1.4 Scheduling { Reparations of oil platforms . .3 1.5 Insurance { Pricing in nonlife insurance . .4 1.6 Power industry { Bidding, power plant operations . .4 1.7 Environment { Inverse modelling in atmosphere . .5 2 Introduction to optimization 6 3 Convex sets and functions 6 4 Separation of sets 8 5 Subdifferentiability and subgradient 9 6 Generalizations of convex functions 10 6.1 Quasiconvex functions . 10 6.2 Pseudoconvex functions . 12 7 Optimality conditions 13 7.1 Optimality conditions based on directions . 13 7.2 Karush{Kuhn{Tucker optimality conditions . 15 7.2.1 A few pieces of the theory . 15 7.2.2 Karush{Kuhn{Tucker optimality conditions . 17 7.2.3 Second Order Sufficient Condition (SOSC) . 19 1 Introduction and motivation 1.1 Operations Research/Management Science and Mathe- matical Programming Goal: improve/stabilize/set of a system. You can reach the goal in the following steps: • Problem understanding • Problem description { probabilistic, statistical and econometric models • Optimization { mathematical programming (formulation and solution) • Verification { backtesting, stresstesting • Implementation (Decision Support
    [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]
  • First Order Characterizations of Pseudoconvex Functions
    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. The second and third characterizations are considered in Sections 5, 6.
    [Show full text]