Subdifferentials of Nonconvex Supremum Functions and Their Applications to Semi-infinite and Infinite Programs with Lipschitzian Data

Abstract

Abstract. The paper is devoted to the subdifferential study and applications of the supremum of uniformly Lipschitzian functions over arbitrary index sets with no topology. Based on advanced techniques of variational analysis, we evaluate major subdifferentials of the supremum functions in the general framework of Asplund (in particular, reflexive) spaces with no convexity or relaxation assumptions. The results obtained are applied to deriving new necessary optimality conditions for nonsmooth and nonconvex problems of semi-infinite and infinite programming.

Slater CQ, optimality and duality for quasiconvex semi-infinite optimization problems

“…This is due to the fact that several engineering problems lead to SIP; see [1,5,14]. Optimality conditions of SIP problems have been studied by many authors; see for instance [1,14] in linear case, [15,19,20,21] in convex case, [5] in smooth case, and [7,8,9,10,29] in locally Lipschitz case.…”

Necessary optimality conditions for countably infinite Lipschitz problems with equality constraint mappings

“…Some of more recent developments in this field are given by Mordukhovich et al in [9][10][11]. These papers mainly address the problem in (2) using different approaches to deriving necessary optimality conditions (not via stability properties of set-valued mappings).…”

“…In [9], the optimality conditions are obtained by computing the normal cones to sets of feasible solutions with differentiable data. The approach in [10] is based on the evaluation of major subdifferentials of the supremum functions.…”

Subdifferential Formulae for the Supremum of an Arbitrary Family of Functions

“…This observation motivates our research to derive general upper estimations for the subdifferential of the supremum function under an arbitrary index set T and without the uniform locally Lipschitz condition. The aim of this work is to extend the results of [28] and give general formulae for the subdifferential of the supremum function, in order to apply them to derive necessary optimality conditions for general problems in the framework of infinite programming. The main motivation for considering an arbitrary family of functions comes from the fact that indicators of sets are commonly used in variational analysis to study constraints and set-valued maps related with optimization problems (for example, stability of optimization problems and differentiability of setvalued maps) and they cannot, at least directly, be assumed to be locally Lipschitz.…”

“…Furthermore, this approach allows us to also study the convex case, and recover general formulae in the convex case, which in particular shows a unifying approach to the study of the subdifferential of the supremum function. For the sake of brevity, we will confine ourselves to extending the results of [28], keeping in mind our applications for a future work.…”