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

Journal of Global Optimization (2022)Cite this article

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.

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Acknowledgements

The work of the first author was completed during his visit to the Shenzhen Research Institute of Big Data, Chinese University of Hong Kong, Shenzhen, China, to which he is grateful to the hospitality received.

Funding

The first author was partially supported by the Chongqing Natural Science Foundation (cstc2021jcyj-msxmX0080), the Postgraduate Research and Innovation Project of Chongqing (2021S0061)” of the second author. the Group Building Scientific Innovation Project for Universities in Chongqing (CXQT21021) and the Education Committee Project Foundation of Bayu Scholar. The study of Yun-Bin Zhao was supported by the National Natural Science Foundation of China (NSFC) under Grant 12071307.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing JiaoTong University, Chongqing, 400074, China

    Zai-Yun Peng, Xue-Jing Chen & Xiao-Bing Li

  2. Shenzhen Research Institute of Big Data, Chinese University of Hong Kong, Shenzhen, 518116, China

    Yun-Bin Zhao

Authors
  1. Zai-Yun Peng
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  2. Xue-Jing Chen
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  3. Yun-Bin Zhao
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  4. Xiao-Bing Li
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Correspondence to Zai-Yun Peng.

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Peng, ZY., Chen, XJ., Zhao, YB. et al. Painlevé-Kuratowski convergence of minimal solutions for set-valued optimization problems via improvement sets. J Glob Optim (2022). https://doi.org/10.1007/s10898-022-01166-8

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  • DOI: https://doi.org/10.1007/s10898-022-01166-8

Keywords

  • Set-valued optimization problem
  • Hausdorff K-convergence
  • Painlevé-Kuratowski convergence
  • Improvement sets

Mathematics Subject Classification

  • 49J40
  • 49K40
  • 90C31