Generic Well-posedness for Perturbed Optimization Problems in Banach Spaces

Let X be a Banach space and Z a relatively weakly compact subset of X. Let J : Z → R be a upper semicontinuous function bounded from above and p ≥ 1. This paper is concerned with the perturbed optimization problem of finding z0 ∈ Z such that ‖x− z0‖ + J(z0) = supz∈Z{‖x− z‖p + J(z)}, which is denoted by maxJ(x, Z). We prove in the present paper that if X is Kadec w.r.t. Z, then the set of all x ∈ X such that the problem maxJ(x, Z) is generalized well-posed is a dense Gδ-subset of X. If X is additionally J-strictly convex w.r.t. Z and p > 1, we prove that the set of all x ∈ X such that the problem maxJ(x, Z) is well-posed is a dense Gδ-subset of X.