Stable generalization of convex functions

“…Tracing back to the original definition of robustly quasiconvex functions, they were first defined in [15] under the name "s-quasiconvex" or "stable quasiconvex", and then renamed "robustly quasiconvex" in [5]. This class of functions holds a notable role, as many important optimization properties of generalized convex functions are stable when disturbed by a linear functional with a sufficiently small norm (for instance, all lower level sets are convex, each minimum is global minimum, each stationary point is a global minimizer).…”

Section: Preliminariesmentioning

confidence: 99%

Characterizations of Nonsmooth Robustly Quasiconvex Functions

Bui

Khanh

Tran

2018

J Optim Theory Appl

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Two criteria for the robust quasiconvexity of lower semicontinuous functions are established in terms of Fréchet subdifferentials in Asplund spaces.The question of characterizing convexity and generalized convexity properties in terms of subdifferentials receives tremendous attention in optimization theory and variational analysis. For decades, there have been received many significant contributions devoted to this question such as [8,9,11,14,16] for convex functions, [2,3,4,6,12,13] for quasiconvex functions and [6] for robustly quasiconvex functions. This paper follows this stream of research. Our aim is to establish the first-order characterizations for the robust quasiconvexity of lower semicontinuous functions in Asplund spaces. First, some existing results regarding to the properties of subdifferential operators of convex, quasiconvex functions are recalled in Section 2, where the definitions and some basic results are given as well. Besides, necessary and sufficient first-order conditions for a lower semicontinuous function to be quasiconvex are reconsidered. Those characterizations moreover could be used to characterize the Asplund property of the given space. Second, two criteria for the robust quasiconvexity of lower semicontinuous functions in Asplund spaces are obtained by using Fréchet subdifferentials in Section 3. Each criterion corresponds to each type of analogous conditions for quasiconvexity. The first one is based on the zero and first order condition for quasiconvexity (see Theorem 2.2(b) in Section 2). It extends [6, Proposition 5.3] from finite dimensional spaces to Asplund spaces. Moreover, its proof also overcomes a glitch in the proof of the sufficient condition of [6, Proposition 5.3]. The second criterion is totally new. It is settled from the equivalence of the quasiconvexity of lower semicontinuous functions and the quasimonotonicity of their subdifferential operators (see Theorem 2.2(c) in Section 2).

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“…It is established in [17] that an equivalent definition is that u is robustly convex (stable convex in the terminology of [17]) if there is an α > 0 such that for any |δ| < α, x 0 , x 1 ∈ Ω, and 0 < λ < 1,…”

Section: Robustly Quasiconvex Functionsmentioning

confidence: 99%

“…We call u : Ω → [−∞, ∞] robustly quasiconvex if it is α-robustly quasiconvex for some α > 0, and denote the class of such functions by R(Ω) = α>0 R α (Ω). We note that robustly quasiconvex functions were named stable-quasiconvex, or just s-quasiconvex, by Phu and An [17]. See [2] and [17] for more examples and discussions of robust quasiconvexity.…”

Section: Introductionmentioning

confidence: 99%