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 $J: Z\rightarrow\mathbb{R}$ be a upper semicontinuous function bounded from above and $p\geq1$. 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 $p>1$, 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 $X$.