Convergence of Solutions of a Set Optimization Problem in the Image Space

Abstract

The present work is devoted to the study of stability in set optimization. In particular, a sequence of perturbed set optimization problems, with a fixed objective map, is studied under suitable continuity assumptions. A formulation of external and internal stability of the solutions is considered in the image space, in such a way that the convergence of a sequence of solutions of perturbed problems to a solution of the original problem is studied under appropriate compactness assumptions. Our results can also be seen as an extension to the set-valued framework of known stability results in vector optimization.

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Acknowledgments

This research was partially supported by the Ministerio de Economía y Competitividad (Spain) under project MTM2012-30942. The third author was also partially supported by MIUR PRIN MISURA Project, 2013-2015, Italy. The authors would like to thank the Editor and the anonymous referee for their useful comments.

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Correspondence to Vicente Novo.

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Communicated by Lionel Thibault.

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Gutiérrez, C., Miglierina, E., Molho, E. et al. Convergence of Solutions of a Set Optimization Problem in the Image Space. J Optim Theory Appl 170, 358–371 (2016). https://doi.org/10.1007/s10957-016-0942-x

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Keywords

  • Set optimization
  • Set relations
  • Minimal solutions
  • Stability
  • Set convergence

Mathematics Subject Classification

  • 49J53
  • 90C31
  • 90C29