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 dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020).
Highlights
Set optimization is the class of mathematical problems that consists in minimizing set-valued mappings acting between two vector spaces, in which the image space is partially ordered by a given closed, convex and pointed cone
Kuroiwa [34] was the first who considered set optimization problems where the solution concept is given by the set approach
We introduce the set relations between the nonempty subsets of Rm that will be used in the definition of the set optimization problem we consider
Summary
Set optimization is the class of mathematical problems that consists in minimizing set-valued mappings acting between two vector spaces, in which the image space is partially ordered by a given closed, convex and pointed cone. The derived algorithms are descent methods and use a derivative-free strategy [7] These algorithms are designed to deal with unconstrained problems, and they assume no particular structure of the set-valued objective mapping. Algorithms based on scalarization [11,12,19,20,27,44] The methods in this group follow a scalarization approach and are derived for problems where the set-valued objective mapping has a particular structure that comes from the so-called robust counterpart of a vector optimization problem under uncertainty, see [20]. The strategy that we consider in this paper is different to the ones previously described and is designed for dealing with unconstrained set optimization problems in which the set-valued objective mapping is given by a finite number of continuously differentiable selections.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
