A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

TL;DR

This paper introduces a first-order steepest descent method for set optimization problems with finite set-valued mappings, providing optimality conditions, convergence analysis, and numerical illustrations.

Contribution

It develops a novel descent algorithm for set optimization with finite-valued mappings and establishes convergence to optimality points.

Findings

The method converges to critical points satisfying optimality conditions.

Numerical examples demonstrate the effectiveness of the proposed descent method.

Abstract

In this paper, we study a first order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems, and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the PhD thesis by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universit\"at Halle-Wittenberg, 2020).