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Chain Rules for a Proper \(\varepsilon \)-Subdifferential of Vector Mappings

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Abstract

In this paper, we derive exact chain rules for a proper epsilon-subdifferential in the sense of Benson of extended vector mappings, recently introduced by ourselves. For this aim, we use a new regularity condition and a new strong epsilon-subdifferential for vector mappings. In particular, we determine chain rules when one of the mappings is linear, obtaining formulations easier to handle in the finite-dimensional case by considering the componentwise order. This Benson proper epsilon-subdifferential generalizes and improves several of the most important proper epsilon-subdifferentials of vector mappings given in the literature and, consequently, the results presented in this work extend known chain rules stated for the last ones. As an application, we derive a characterization of approximate Benson proper solutions of implicitly constrained convex Pareto problems. Moreover, we estimate the distance between the objective values of these approximate proper solutions and the set of nondominated attained values.

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Acknowledgments

This work was partially supported by Ministerio de Economía y Competitividad (Spain) under project MTM2012-30942. The authors are very grateful to the anonymous referee for his/her helpful comments and suggestions.

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Authors and Affiliations

  1. Universidad de Valladolid, Valladolid, Spain

    César Gutiérrez

  2. Universidad Nacional de Educación a Distancia, Madrid, Spain

    Lidia Huerga & Vicente Novo

  3. Université Montpellier 2, Montpellier, France

    Lionel Thibault

Authors
  1. César Gutiérrez
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  2. Lidia Huerga
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  3. Vicente Novo
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  4. Lionel Thibault
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Corresponding author

Correspondence to Vicente Novo.

Additional information

Communicated by Boris S. Mordukhovich.

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Gutiérrez, C., Huerga, L., Novo, V. et al. Chain Rules for a Proper \(\varepsilon \)-Subdifferential of Vector Mappings. J Optim Theory Appl 167, 502–526 (2015). https://doi.org/10.1007/s10957-015-0763-3

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  • DOI: https://doi.org/10.1007/s10957-015-0763-3

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