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

arXiv: https://arxiv.org/abs/2107.12122 · HTML: https://mlsanity.com/paper/2107.12122/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jul 2021.

## Built on by (most cited first, 1 of 1)

- [Stochastic set-valued optimization and its application to robust learning](https://mlsanity.com/paper/2603.17691/) (2026)

## Abstract (arXiv)

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).