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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. 2Assistant Professor, Mathematiques Appliquees, Universite de Perpignan, Perpignan, France. 29 0022-3239/98/0400-0029$15.00/0 © 1998 Plenum Publishing Corporation 30 JOTA: VOL. 97, NO. 1, APRIL 1998 The paper is organized as follows. Section 2 is devoted to the study of quasiconvex functions. We establish a characterization based on a mixed property combining properties of the function and properties of its subdiff- erential. The case of quasiconcave and quasiaffine functions, for which anal- ogous characterizations are obtained, is treated in Section 3. In Section 4, we generalize to the nonsmooth setting a result of Crouzeix and Ferland (Ref. 2) which describes the precise relation between quasiconvexity and pseudoconvexity. Moreover, this allows one to prove in an elegant way two characterizations of pseudoconvexity. Finally, we point out in Section 5 that the continuity assumption can be replaced in some results by the radially nonconstant hypothesis, already used by Diewert in 1981 (Ref. 3). In all the sequel, X denotes a Banach space, X* its topological dual, and < •, • > the duality pairing. Most functionsf: X-+R u { + 00} considered in this paper will be (at least) lower semicontinuous (l.s.c. in short), and dom f will denote the domain off For a point-to-set map A: X-+X*, we let Throughout this paper, d stands for the following abstract notion of subdifferential. Definition 1.1. See Ref. 1. We call subdifferential, denoted by d, any operator which associates a subset Sf(x) of X* to any lower semicontinuous f: X-**R u {+00} and any xeX, and satisfies the following properties: (P1) df(x) = {x*eX*\<x*,y-xy+f(x)<f(y), VyeX}, whenever f is convex; (P2) Oedf(x), whenever xedom f is a local minimum of f; (P3) d(f+g)(x)<df(x) + 8g(x), whenever g is a real-valued convex continuous function which is 3-differentiable at x. Here, g is d-differentiable at x means that both dg(x) and d(-g)(x) are nonempty. We say that a function f is 3-subdifferentiable at x when df(x) is nonempty. This abstract subdifferential allows one to recover, by a unique defini- tion, a large class of subdifferentials. Among this class, let us mention some examples (for precise definitions, see Ref. 1 and references therein): the Clarke-Rockafellar subdifferential dCRf; the lower and upper Dini subdifferentials <3D- f and <3D+ f; JOTA: VOL. 97, NO. 1, APRIL 1998 31 the lower Hadamard subdifferential dH- f, also called contingent subdifferential; the Frechet subdifferential dF f; the Lipschitz smooth subdifferential dLS f, also called proximal subdiff- erential d" f whenever X is a Hilbert space. The verification that these subdifferentials satisfy properties (P1) to (P3) is straightforward (see Ref. 4) and does not require any assumption on the Banach space. The use of the word d-differentiable is justified by the follow- ing observation: a function is <3°~-differentiable [resp. dH--differentiable, dF-differentiable] at x if and only if it is Gateaux [resp. Hadamard, Frechet] differentiable at x. In all the sequel, we will focus our attention on the class of subdifferen- tials which satisfy properties (P1) to (P3) and one of the following inclusions: This assumption is not needed in all the proofs and in any case is not really restrictive, since the Clarke-Rockafellar and the upper Dini subdifferentials are the biggest (in the sense of inclusion) among the classical subdifferentials. In particular, one has Let us recall the definition of the Clarke-Rockafellar subdifferential and the definition of the upper Dini subdifferential which will be helpful for the sequel, (one can take «=f(u) whenever f is l.s.c.); 32 JOTA: VOL. 97, NO. 1, APRIL 1998 Definition 1.2. See Ref. 1.3 A norm || • || on X is said to be d-smooth if all the (real-valued, convex, continuous) functions of the following form are 3-differentiable: (i) df[a,b](x) •= min,6[a,fc] ||x-c||2, where [a, b] is a closed segment in X, (ii) A2(x):=I.nnn\\x-vn\\2, where 2nJuM= 1, ^«>0, and (i>«) converges in X. We say that a Banach space admits a d-smooth renorm if it admits an equivalent norm which is 3-smooth. Let us give some interesting examples of such d-smooth norms: (a) a norm is ^"-smooth iff it is dfl~-differentiable on ^f\{0}, that is, according to a previous remark, iff it is Gateaux-differentiable on X\{0] (elementary proof); (b) any norm is dc/J-smooth since the functions rff0ifc] and A2 are locally Lipschitz. Concerning (a), we note that the same equivalence, with respect to Hadamard or Frechet differentiability, is true for d — dH- or d = dF. So, any separable [resp. reflexive] Banach space admits a <3D--smooth [resp. dF- smooth] renorm; see, e.g., Ref. 6. Finally, we need the following result which is a special case of the mean- value inequality stated in Ref. 1. Proposition 1.1. Let X be a Banach space with a 3-smooth renorm, and let f:X-»R u {+00} be a l.s.c. function. For any aedom f and beX such that f(a)<f(b), there exist ce[a, b[ and sequences (xn) converging to c and (x*), x*edf(xn), such that for every d=c + t(b-a) with t>0. 2. Two Characterizations of Quasiconvexity This section is devoted to the study of quasiconvex functions. More precisely, we extract from the proof of the main result of Aussel et al. (Ref. 7, Theorem 1) a mixed property (Qs) combining properties of the function 3Since this paper was submitted, J. N. Corvellec, M. Lassonde, and the author have shown that (i) can be deduced from (ii); see Proposition 2.3 in Ref. 5. JOTA: VOL. 97, NO. 1, APRIL 1998 33 and properties of the subdifferential. Using the same arguments as in Ref. 7, Theorem 1, we show that this mixed property characterizes the quasi- convexity of l.s.c. functions. This characterization will play a central role in the sequel. Let us recall that a function is quasiconvex if, for any x, yeX and any ze[x,y], one has It is well known that, in the differentiable case, quasiconvex functions satisfy In the nondifferentiable case, this property becomes Our first result asserts that the following slightly stronger mixed property actually characterizes the quasiconvexity of l.s.c. functions: Theorem 2.1. Let X be a Banach space with a 5-smooth renorm, and let f: X ->R u { + 00} be a l.s.c. function. Then, the following assertions are equivalent: (i) f is quasiconvex; Proof. (i) =» (ii) Case dfcdCRf. Let x,yedom df and x*edf(x) such that Hence, there exists e > 0 such that, for every ne N*, one can find xneBe/n(x) [and then y — xneBe(y — x)] and tne ]0, 1/n[ which satisfy Since f is quasiconvex, the previous inequality implies that, for every te [0, 1 ], we have and hence, by lower semicontinuity of f, (i) => (ii) Case df c BD+f. The proof is very simple. Indeed, if x, y and x*edD+f(x) satisfy fD+(x,y-x)>0, then f(z)>f(x) for some 34 JOTA: VOL. 97, NO. 1, APRIL 1998 ze ]x, y[, and by quasiconvexity of f, (ii) => (i) We essentially reproduce the first part of the proof of Ref. 7, Theorem 1. Let x, yedom f and According to Proposition 1.1, there exist sequences (xn) and (x*) such that xn->xe[x, z[, x*edf(xn), and <x*,y-xn>>0, VneM. Hypothesis (ii) implies that, for every neN, the point zn defined on ]xn,y] by zn = Axn + (l-A)y satisfies f(zn)<f(y), and hence by lower semicontinuity f(z)<f(y). D Obviously, every l.s.c. quasiconvex function also satisfies property (0. But the converse is false, in general, for l.s.c.
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