Solving set-valued optimization problems using a multiobjective approach

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.

KEYWORDS:

• Set-valued optimization
• minimal value function
• multiobjective optimization
• approximation algorithms

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 90C48
• 90C29
• 90C59

Acknowledgments

The authors would like to thank Ernest Quintana from TU Ilmenau, who contributed with fruitful discussions to the proof of Theorem 3.12. The authors would also like to thank the two anonymous referees whose suggestions and remarks helped to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of the second author was funded by DFG [grant number 392195690].