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

Gemayqzel Bouza (), Ernest Quintana () and Christiane Tammer ()
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Gemayqzel Bouza: University of Havana
Ernest Quintana: Technical University of Ilmenau
Christiane Tammer: Martin-Luther University of Halle-Wittenberg

Journal of Optimization Theory and Applications, 2021, vol. 190, issue 3, No 1, 743 pages

Abstract: 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).

Keywords: Set optimization; Robust vector optimization; Descent method; Stationary point; 49J53; 90C29; 90C46; 90C47 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-021-01887-y

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