Well-posedness of a class of perturbed optimization problems in Banach spaces

Abstract

Let X be a Banach space and Z a nonempty subset of X. Let J : Z → R be a lower semicontinuous function bounded from below and p 1. This paper is concerned with the perturbed optimization problem of findingThe notions of the J -strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x ∈ X for which every minimizing sequence of the problem min J (x, Z ) has a converging subsequence is a dense G δ -subset of X \ Z 0 , where Z 0 is the set of all points z ∈ Z such that z is a solution of the problem min J (z, Z ). If additionally p > 1 and X is J -strictly convex with respect to Z , then the set of all x ∈ X for which the problem min J (x, Z ) is well-posed is a dense G δ -subset of X \ Z 0 .

Levitin-Polyak Well-Posedness for Generalized Quasi-Variational Inclusion and Disclusion Problems and Optimization Problems With Constraints

“…Well-posedness of unconstrained and constrained scalar optimization problems was first introduced and studied by Tykhonov [45] and Levitin and Polyak [21], respectively. Since then, various concepts of well-posedness have been introduced and extensively studied for minimization problems and vector optimization problems by many authors (see, for example, [2,6,7,15,38,40,42,47,48] and the references therein).…”

Generic uniqueness of equilibrium points

“…Since Q is a second category set, we may say that the property P holds for most of points (in the sense of Baire category) in X . The research on generic properties (including generic existence, generic uniqueness, generic stability, generic well-posedness and so on) has attracted much attention; see [2][3][4][5][6][7][8][9][10][11][12][16][17][18] and the references therein.…”

Nearest and farthest points in spaces of curvature bounded below

“…For more developments and extensions in this direction, the readers are referred to [2,3,7,8,11,[22][23][24][25][26][27][28][29][31][32][33] and the surveys [12,30].…”