Abstract
We study the stability of set-valued optimization problems by using epi-convergence coupled with asymptotic analysis. To do this, we recall the notion of epi-convergence and introduce the notion of total epi-convergence for set-valued maps. We characterize them by means of epi-limits and horizon epi-limits. We use these epi-limits to study the behavior of vector/set type solutions and level/colevel sets under variations of the whole data. We also introduce several stronger epi-convergence notions and use them to study the stability of solution sets and minimal solution sets. We extend and generalize various results from the literature since we deal with unbounded feasible sets and objective maps. Finally, to illustrate our main results, we apply them to various classes of set-valued maps.
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The authors are very grateful to the anonymous referee for his/her helpful and valuable comments and suggestions.
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This research was funded by the Spanish Ministerio de Economía y Competitividad and FEDER, Grant MTM2017-85054-C2-2-P and ETSI Industriales UNED, Project 2022-ETSII-UNED-16 (Hernández) and by ANID–Chile under project Fondecyt 1181368 (López)
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Hernández, E., López, R. Stability in Set-Valued Optimization Problems Using Asymptotic Analysis and Epi-Convergence. Appl Math Optim 86, 7 (2022). https://doi.org/10.1007/s00245-022-09879-8
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DOI: https://doi.org/10.1007/s00245-022-09879-8
Keywords
- Set-valued map
- Set-valued optimization problem
- Asymptotic map
- Epi-convergence
- Total epi-convergence
- Stability