DOCSLIB.ORG

Sum of Squares and Polynomial Convexity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. INTRODUCTION measures. In a different work [8], Lasserre appeals to sos- In many problems in applied and computational math- convexity to give sufficient conditions for a polynomial ematics, we would like to decide whether a multivariate to belong to the quadratic module generated by a set of polynomial is convex or to parameterize a family of convex polynomials. More on the practical side, Magnani, Lall, and polynomials. Perhaps the most obvious instance appears Boyd [9] use sum of squares programming to find sos-convex in optimization. It is well known that in the absence of polynomials that best fit a set of data points or to find convexity, global minimization of polynomials is generally minimum volume convex sets, given by sub-level sets of sos- NP-hard [1], [2], [3]. However, if we somehow know a priori convex polynomials, that contain a set of points in space. that the polynomial is convex, nonexistence of local minima Even though it is well-known that sum of squares and is guaranteed and simple gradient descent methods can find a nonnegativity are not equivalent, because of the special global minimum. In many other practical settings, we might structure of the Hessian matrix, sos-convexity and convexity want to parameterize a family of convex polynomials that could potentially turn out to be equivalent. This speculation have certain properties, e.g., that serve as a convex envelope has been bolstered by the fact that finding a counterexample for a non-convex function, approximate a more complicated has shown to be difficult and attempts at giving a non- function, or fit some data points with minimum error. To constructive proof of its existence have seen no success address many questions of this type, we need to have an either. Our contribution in this paper is to give the first understanding of the algebraic structure of the set of convex such counterexample, i.e., the first example of a polynomial polynomials. that is convex but not sos-convex. This example is presented Over a decade ago, Pardalos and Vavasis [4] put the in Theorem 3.2. Our result further supports the hypothesis following question proposed by Shor on the list of seven that deciding convexity of polynomials should be a difficult most important open problems in complexity theory for problem. We hope that our counterexample, in a similar way numerical optimization: “Given a degree-4 polynomial in n to what other celebrated counterexamples [10]–[13] have variables, what is the complexity of determining whether this achieved, will help stimulate further research and clarify the polynomial describes a convex function?” To the best of our relationships between the geometric and algebraic aspects of knowledge, the question is still open but the general belief positivity and convexity. is that the problem should be hard (see the related work in The organization of the paper is as follows. Section II [5]). Not surprisingly, if testing membership to the set of is devoted to mathematical preliminaries required for under- convex polynomials is hard, searching and optimizing over standing the remainder of this paper. We begin this section them also turns out to be a hard problem. by introducing the cone of nonnegative and sum of squares The notion of sos-convexity has recently been proposed polynomials. We briefly discuss the connection between as a tractable relaxation for convexity based on semidefinite sum of squares decomposition and semidefinite programming programming. Broadly speaking, the requirement of positive highlighting also the dual problem. Formal definitions of sos- convex polynomials and sos-matrices are also given in this Amir Ali Ahmadi and Pablo A. Parrilo are with the Laboratory for section. In Section III, we present our main result, which is Information and Decision Systems, Department of Electrical Engineering and Computer Science at Massachusetts Institute of Technology. E-mail: an explicit example of a convex polynomial that is not sos- a a [email protected], [email protected]. convex. Some of the properties of this polynomial are also Preprint submitted to 48th IEEE Conference on Decision and Control. Received March 6, 2009. CONFIDENTIAL. Limited circulation. For review only. discussed at the end of this section. n variables can be homogenized into a form ph of degree d A more comprehensive version of this paper is presented in n + 1 variables, by adding a new variable y, and letting in [14]. Because our space is limited here, some of the x x p (x ; : : : ; x ; y) := ydp( 1 ;:::; n ): technical details and parts of the proofs have been omitted h 1 n y y and can be found in [14]. Furthermore, in [14], we explain The properties of being psd and sos are preserved under how we have found our counterexample using software and homogenization and dehomogenization. by utilizing techniques from sum of squares programming To make the ideas presented so far more concrete, we end and duality theory of semidefinite optimization. The method- this section by discussing the example of Motzkin. ology explained there is of independent interest since it can Example 2.1: The Motzkin polynomial be employed to search or optimize over a restricted family 4 2 2 4 2 2 of nonnegative polynomials that are not sums of squares. M(x1; x2) = x1x2 + x1x2 − 3x1x2 + 1 (2) II. MATHEMATICAL BACKGROUND is historically the first known example of a polynomial that A. Nonnegativity and sum of squares is psd but not sos. Positive semidefiniteness follows from the arithmetic-geometric inequality, and the nonexistence of an We denote by K[x] := K[x1; : : : ; xn] the ring of poly- sos decomposition can be shown by some clever algebraic nomials in n variables with coefficients in the field K. manipulations (see [16]) or by a duality argument. We can Throughout the paper, we will have K = R or K = Q.A homogenize this polynomial and obtain the Motzkin form polynomial p(x) 2 R[x] is said to be nonnegative or positive n 4 2 2 4 2 2 2 6 semidefinite (psd) if p(x) ≥ 0 for all x 2 R . Clearly, a Mh(x1; x2; x3) := x1x2 + x1x2 − 3x1x2x3 + x3; (3) necessary condition for a polynomial to be psd is for its total which belongs to P3;6 n Σ3;6 as expected. degree to be even. We say that p(x) is a sum of squares (sos), if there exist polynomials q1(x); :::; qm(x) such that B. Sum of squares, semidefinite programming, and duality m Deciding nonnegativity of polynomials is an important X 2 p(x) = qi (x): (1) problem that arises in many areas of systems theory, control, i=1 and optimization [17]. Unfortunately, this problem is known It is clear that p(x) being sos implies that p(x) is psd. In to be NP-hard even when the degree of the polynomial is 1888, David Hilbert [15] proved that the converse is true equal to four [2], [3]. On the other hand, deciding whether for a polynomial in n variables and of degree d only in the a given polynomial admits an sos decomposition turns out following cases: to be a tractable problem. This tractability stems from the underlying convexity of the problem as first pointed out in • n = 1 (univariate polynomials of any degree) [18], [19], [20]. More specifically, it was shown in [20] • d = 2 (quadratic polynomials in any number of vari- that one can reduce the problem of deciding whether a ables) polynomial is sos to feasibility of a semidefinite program • n = 2; d = 4 (bivariate quartics) (SDP). Semidefinite programs are a well-studied subclass of Hilbert showed that in all other cases there exist poly- convex optimization problems that can be efficiently solved nomials that are psd but not sos. Explicit examples of in polynomial time using interior point algorithms. Because such polynomials appeared nearly 80 years later, starting our space is limited, we refrain from further discussing with the celebrated example of Motzkin, followed by more SDPs and refer the interested reader to the excellent review examples by Robinson, Choi and Lam, and Lax-Lax and paper [21].
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]
  • 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]
  • Convexity I: Sets and Functions
    Convexity I: Sets and Functions Ryan Tibshirani Convex Optimization 10-725/36-725 See supplements for reviews of basic real analysis • basic multivariate calculus • basic linear algebra • Last time: why convexity? Why convexity? Simply put: because we can broadly understand and solve convex optimization problems Nonconvex problems are mostly treated on a case by case basis Reminder: a convex optimization problem is of ● the form ● ● min f(x) x2D ● ● subject to gi(x) 0; i = 1; : : : m ≤ hj(x) = 0; j = 1; : : : r ● ● where f and gi, i = 1; : : : m are all convex, and ● hj, j = 1; : : : r are affine. Special property: any ● local minimizer is a global minimizer ● 2 Outline Today: Convex sets • Examples • Key properties • Operations preserving convexity • Same for convex functions • 3 Convex sets n Convex set: C R such that ⊆ x; y C = tx + (1 t)y C for all 0 t 1 2 ) − 2 ≤ ≤ In words, line segment joining any two elements lies entirely in set 24 2 Convex sets Figure 2.2 Some simple convexn and nonconvex sets. Left. The hexagon, Convex combinationwhich includesof x1; its : :boundary : xk (shownR : darker), any linear is convex. combinationMiddle. The kidney shaped set is not convex, since2 the line segment between the twopointsin the set shown as dots is not contained in the set. Right. The square contains some boundaryθ1x points1 + but::: not+ others,θkxk and is not convex. k with θi 0, i = 1; : : : k, and θi = 1. Convex hull of a set C, ≥ i=1 conv(C), is all convex combinations of elements. Always convex P 4 Figure 2.3 The convex hulls of two sets in R2.
    [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]
  • 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.
    [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]
  • Convex Functions
    Convex Functions Prof. Daniel P. Palomar ELEC5470/IEDA6100A - Convex Optimization The Hong Kong University of Science and Technology (HKUST) Fall 2020-21 Outline of Lecture • Definition convex function • Examples on R, Rn, and Rn×n • Restriction of a convex function to a line • First- and second-order conditions • Operations that preserve convexity • Quasi-convexity, log-convexity, and convexity w.r.t. generalized inequalities (Acknowledgement to Stephen Boyd for material for this lecture.) Daniel P. Palomar 1 Definition of Convex Function • A function f : Rn −! R is said to be convex if the domain, dom f, is convex and for any x; y 2 dom f and 0 ≤ θ ≤ 1, f (θx + (1 − θ) y) ≤ θf (x) + (1 − θ) f (y) : • f is strictly convex if the inequality is strict for 0 < θ < 1. • f is concave if −f is convex. Daniel P. Palomar 2 Examples on R Convex functions: • affine: ax + b on R • powers of absolute value: jxjp on R, for p ≥ 1 (e.g., jxj) p 2 • powers: x on R++, for p ≥ 1 or p ≤ 0 (e.g., x ) • exponential: eax on R • negative entropy: x log x on R++ Concave functions: • affine: ax + b on R p • powers: x on R++, for 0 ≤ p ≤ 1 • logarithm: log x on R++ Daniel P. Palomar 3 Examples on Rn • Affine functions f (x) = aT x + b are convex and concave on Rn. n • Norms kxk are convex on R (e.g., kxk1, kxk1, kxk2). • Quadratic functions f (x) = xT P x + 2qT x + r are convex Rn if and only if P 0.
    [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]
  • Subgradients
    Subgradients Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: gradient descent Consider the problem min f(x) x n for f convex and differentiable, dom(f) = R . Gradient descent: (0) n choose initial x 2 R , repeat (k) (k−1) (k−1) x = x − tk · rf(x ); k = 1; 2; 3;::: Step sizes tk chosen to be fixed and small, or by backtracking line search If rf Lipschitz, gradient descent has convergence rate O(1/) Downsides: • Requires f differentiable next lecture • Can be slow to converge two lectures from now 2 Outline Today: crucial mathematical underpinnings! • Subgradients • Examples • Subgradient rules • Optimality characterizations 3 Subgradients Remember that for convex and differentiable f, f(y) ≥ f(x) + rf(x)T (y − x) for all x; y I.e., linear approximation always underestimates f n A subgradient of a convex function f at x is any g 2 R such that f(y) ≥ f(x) + gT (y − x) for all y • Always exists • If f differentiable at x, then g = rf(x) uniquely • Actually, same definition works for nonconvex f (however, subgradients need not exist) 4 Examples of subgradients Consider f : R ! R, f(x) = jxj 2.0 1.5 1.0 f(x) 0.5 0.0 −0.5 −2 −1 0 1 2 x • For x 6= 0, unique subgradient g = sign(x) • For x = 0, subgradient g is any element of [−1; 1] 5 n Consider f : R ! R, f(x) = kxk2 f(x) x2 x1 • For x 6= 0, unique subgradient g = x=kxk2 • For x = 0, subgradient g is any element of fz : kzk2 ≤ 1g 6 n Consider f : R ! R, f(x) = kxk1 f(x) x2 x1 • For xi 6= 0, unique ith component gi = sign(xi) • For xi = 0, ith component gi is any element of [−1; 1] 7 n
    [Show full text]
  • NP-Hardness of Deciding Convexity of Quartic Polynomials and Related Problems
    NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems Amir Ali Ahmadi, Alex Olshevsky, Pablo A. Parrilo, and John N. Tsitsiklis ∗y Abstract We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time. 1 Introduction The role of convexity in modern day mathematical programming has proven to be remarkably fundamental, to the point that tractability of an optimization problem is nowadays assessed, more often than not, by whether or not the problem benefits from some sort of underlying convexity. In the famous words of Rockafellar [39]: \In fact the great watershed in optimization isn't between linearity and nonlinearity, but convexity and nonconvexity." But how easy is it to distinguish between convexity and nonconvexity? Can we decide in an efficient manner if a given optimization problem is convex? A class of optimization problems that allow for a rigorous study of this question from a com- putational complexity viewpoint is the class of polynomial optimization problems. These are op- timization problems where the objective is given by a polynomial function and the feasible set is described by polynomial inequalities.
    [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]