Genericity of well-posed vector optimization problems

TL;DR

This paper extends the concept of generic well-posedness from scalar to vector optimization problems in Banach spaces, showing that such well-posed problems are common or 'generic' under certain conditions.

Contribution

It generalizes existing scalar optimization genericity results to vector optimization problems, establishing the prevalence of well-posed problems in this broader context.

Findings

Well-posed vector optimization problems form a dense set under certain conditions.

The set of well-posed problems contains a dense G_delta set, indicating genericity.

Results extend scalar optimization genericity to vector optimization in Banach spaces.

Abstract

In this paper we consider well-posedness properties of vector optimization problems with objective function $f: X \to Y$ where $X$ and $Y$ are Banach spaces and $Y$ is partially ordered by a closed convex pointed cone with nonempty interior. The vector well-posedness notion considered in this paper is the one due to Dentcheva and Helbig, which is a natural extension of Tykhonov well-posedness for scalar optimization problems. When a scalar optimization problem is considered it is possible to prove that under some assumptions the set of functions for which the related optimzation problem is well-posed is dense or even more in "big" i.e. contains a dense $G_{\delta}$ set (these results are called genericity results). The aim of this paper is to extend these genericity results to vector optimization problems.