A Vectorization Scheme for Nonconvex Set Optimization Problems

TL;DR

This paper introduces a vectorization scheme for nonconvex set optimization problems, transforming them into parametric multiobjective problems to facilitate numerical solutions with established solvers.

Contribution

It proposes a novel approach to approximate set optimization problems via a parametric family of multiobjective problems, including cases with convex graphs.

Findings

The scheme effectively approximates solutions of nonconvex set optimization problems.

Certain classes of set-valued mappings with convex graphs are equivalent to multiobjective problems.

The approach enables the use of existing multiobjective solvers for set optimization.

Abstract

In this paper, we study a solution approach for set optimization problems with respect to the lower set less relation. This approach can serve as a base for numerically solving set optimization problems by using established solvers from multiobjective optimization. Our strategy consists of deriving a parametric family of multiobjective optimization problems whose optimal solution sets approximate, in a specific sense, that of the set-valued problem with arbitrary accuracy. We also examine particular classes of set-valued mappings for which the corresponding set optimization problem is equivalent to a multiobjective optimization problem in the generated family. Surprisingly, this includes set-valued mappings with a convex graph.