On some global implicit function theorems for set-valued inclusions with applications to parametric vector optimization

TL;DR

This paper develops global implicit function theorems for set-valued inclusions, enabling analysis of solution stability and existence in parametric vector optimization problems using variational analysis techniques.

Contribution

It introduces new qualitative global implicit function theorems for set-valued inclusions with parameters, ensuring solution existence and continuous dependence, with applications to vector optimization.

Findings

Established global implicit function theorems for set-valued inclusions.

Proved continuous dependence of solutions on parameters.

Provided sufficient conditions for existence of ideal efficient solutions.

Abstract

The present paper deals with the perturbation analysis of set-valued inclusion problems, a problem format whose relevance has recently emerged in such contexts as robust and vector optimization as well as in vector equilibrium theory. The set-valued inclusions here considered are parameterized by variables belonging to a topological space, with and without constraints. By proper techniques of variational analysis, some qualitative global implicit function theorems are established, which ensure global solvability of these problems and continuous dependence on the parameter of the related solutions. Applications to parametric vector optimization are discussed, aimed at deriving sufficient conditions for the existence of ideal efficient solutions that depend continuously on the parameter perturbations.