GENERIC WELL-POSEDNESS FOR PERTURBED OPTIMIZATION PROBLEMS IN BANACH SPACES

Abstract

Let $X$ be a Banach space and $Z$ a relatively weakly compact subset of $X$. Let $J: Z \rightarrow \mathbb{R}$ be a upper semicontinuous function bounded from above and $p \geq 1$. This paper is concerned with the perturbed optimization problem of finding $z_0 \in Z$ such that $\|x-z_0\|^p + J(z_0) = \sup_{z \in Z} \{\|x-z\|^p + J(z)\}$, which is denoted by $\max_J(x,Z)$. We prove in the present paper that if $X$ is Kadec w.r.t. $Z$, then the set of all $x \in X$ such that the problem $\max_J(x,Z)$ is generalized well-posed is a dense $G_\delta$-subset of $X$. If $X$ is additionally $J$-strictly convex w.r.t. $Z$ and $p \gt 1$, we prove that the set of all $x\in X$ such that the problem $\max_J(x,Z)$ is well-posed is a dense $G_\delta$-subset of $X$.