Painlevé-Kuratowski convergence of minimal solutions for set-valued optimization problems via improvement sets

Zai-Yun Peng (), Xue-Jing Chen (), Yun-Bin Zhao () and Xiao-Bing Li ()
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Zai-Yun Peng: Chongqing JiaoTong University
Xue-Jing Chen: Chongqing JiaoTong University
Yun-Bin Zhao: Chinese University of Hong Kong
Xiao-Bing Li: Chongqing JiaoTong University

Journal of Global Optimization, 2023, vol. 87, issue 2, No 19, 759-781

Abstract: Abstract The aim of this paper is to explore the stability of (weak)-minimal solutions for set-valued optimization problems via improvement sets. Firstly, the optimality and closedness of solution sets for the set-valued optimization problem under the upper order relation are discussed. Then, a new convergence concept for set-valued mapping sequences is introduced, and some properties of the set-valued mapping sequences are shown under the new convergence assumption. Moreover, by means of upper level sets, Painlevé-Kuratowski convergences of (weak) E-u-solutions to set-valued optimization problems with respect to the perturbations of feasible sets and objective mappings are established under mild conditions. The order that we use to establish the result depends on the improvement set, which is not necessarily a cone order. Our results can be seen as the extension of the related work established recently in this field.

Keywords: Set-valued optimization problem; Hausdorff K-convergence; Painlevé-Kuratowski convergence; Improvement sets; 49J40; 49K40; 90C31 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10898-022-01166-8

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