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Scalarization, convergence and well-posedness in set optimization

Taiyong Li & Manli Yang

Journal of Industrial and Management Optimization2026https://doi.org/10.3934/jimo.2026044article
AJG 1ABDC B
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0.50

Abstract

In this paper, we introduce the notions of weak $ m $-minimal approximate solutions and $ m $-minimal approximate solutions for constrained set optimization problems, based on a novel set order relation that involves the Minkowski difference. We derive scalarization results for the sets of weak $ m $-minimal approximate solutions and $ m $-minimal approximate solutions in the context of set optimization. Utilizing these scalarizations, we analyze the Painlevé-Kuratowski convergence properties of both classes of approximate solutions. Furthermore, new notions of well-posedness for set optimization problems are proposed, and relationships between these notions are rigorously established. Finally, we establish equivalences between the well-posedness of set optimization problems and their scalar counterparts through carefully constructed optimization frameworks.

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https://doi.org/https://doi.org/10.3934/jimo.2026044

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@article{taiyong2026,
  title        = {{Scalarization, convergence and well-posedness in set optimization}},
  author       = {Taiyong Li & Manli Yang},
  journal      = {Journal of Industrial and Management Optimization},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.3934/jimo.2026044},
}

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