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

Author

Listed:
  • Gemayqzel Bouza

    (University of Havana)

  • Ernest Quintana

    (Technical University of Ilmenau)

  • Christiane Tammer

    (Martin-Luther University of Halle-Wittenberg)

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).

Suggested Citation

  • Gemayqzel Bouza & Ernest Quintana & Christiane Tammer, 2021. "A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 711-743, September.
    • Handle: RePEc:spr:joptap:v:190:y:2021:i:3:d:10.1007_s10957-021-01887-y
    DOI: 10.1007/s10957-021-01887-y
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    References listed on IDEAS

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