Solving set-valued optimization problems using a multiobjective approach. (4th March 2023)

• Record Type: Journal Article
• Authors: Eichfelder, Gabriele; Rocktäschel, Stefan
• Is Part Of: Optimization. Volume 72:Number 3(2023)
• Page Start: 789
• Page End: 820
• Publication Date: 2023-03-04
• Subjects: Set-valued optimization -- minimal value function -- multiobjective optimization -- approximation algorithms; 90C48 -- 90C29 -- 90C59; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2021.1988596
• Languages: English
• ISSNs: 0233-1934

Abstract

ABSTRACT: Set-valued optimization using the set approach is a research topic of high interest due to its practical relevance and numerous interdependencies to other fields of optimization. However, it is a very difficult task to solve these optimization problems even for specific cases. In this paper, we study set-valued optimization problems and develop a multiobjective optimization problem that is strongly related to it. We prove that the set of weakly minimal solutions of this subproblem is closely related to the set of weakly minimal elements of the set-valued optimization problem and that these sets can get arbitrarily close in a certain sense. Subsequently, we introduce a concept of approximations of the solution set of the set-valued optimization problem. We define a quality measure in the image space that can be used to compare finite approximations of this kind and outline a procedure to enhance a given approximation. We conclude the paper with some numerical examples.