Unified approach to some geometric results in variational analysis

Abstract

Based on a study of a minimization problem, we present the following results applicable to possibly nonconvex sets in a Banach space: an approximate projection result, an extended extremal principle, a nonconvex separation theorem, a generalized Bishop-Phelps theorem and a separable point result. The classical result of Dieudonné (on separation of two convex sets in a finite-dimensional space) is also extended to a nonconvex setting.

A Unified Separation Theorem for Closed Sets in a Banach Space and Optimality Conditions for Vector Optimization

“…Fuzzy separation results. In this section, we establish fuzzy separation results for finitely many closed sets, which not only unifies the convex separation theorem mentioned in section 1 and the existing nonconvex separation results, but also recaptures the approximate projection theorem proved in [30] and [12].…”

“…Then, for any λ > 0, there existā i ∈ A i and a à i ∈ X à such that (i) P n i¼1 kā i − a i k < λ, max 1≤i≤n−1 ka à i k ¼ 1, and P n i¼1 a Unfortunately, even in the case when n ¼ 2, A 1 ¼ fxg, and A 2 is convex (and closed) such that x ∈ = A 2 , this theorem and all other existing fuzzy separation results for general closed sets cannot recapture the classical separation theorem stated at the beginning of this section. On the other hand, by the approximate projection theorem for a closed set (proved by the authors [30] and [12]), for any η ∈ ð0; 1Þ, there existā 2 ∈ A 2 and −a Clearly, (1.2) does imply that A 1 ¼ fxg and A 2 can be separated (in the usual sense) if A 2 is convex. From the theoretical viewpoint as well as for applications, it is important and interesting to have a new kind of fuzzy separation theorem that can result in existing fuzzy separation theorems and classical convex separation results.…”

Metric Subregularity of Multifunctions: First and Second Order Infinitesimal Characterizations

“…Let X be a Banach space. We use the symbol to denote any abstract subdifferential, that is, any set-valued mapping that associates to every function defined on X and every x ∈ X, a set f x ⊂ X (possibly empty), in such a way that (see Aussel et al [3], Lassonde [36], Li et al [39], Mordukhovich [41,42,43]), Penot [51].…”

Extremality, Stationarity and Generalized Separation of Collections of Sets

“…Observe that conditions (29) and (30) are exactly conditions (P5) and (P2), respectively, applied to the localizations of the sets Ω 1 −ω 1 , . .…”

“…and the expressions involved in the first condition in(29) correspond to localizations of the sets Ω i −ω i (i = 1, . .…”