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Chain Rules for a Proper $$\varepsilon $$ ε -Subdifferential of Vector Mappings

Author

Listed:
  • César Gutiérrez

    (Universidad de Valladolid)

  • Lidia Huerga

    (Universidad Nacional de Educación a Distancia)

  • Vicente Novo

    (Universidad Nacional de Educación a Distancia)

  • Lionel Thibault

    (Université Montpellier 2)

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.

Suggested Citation

  • César Gutiérrez & Lidia Huerga & Vicente Novo & Lionel Thibault, 2015. "Chain Rules for a Proper $$\varepsilon $$ ε -Subdifferential of Vector Mappings," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 502-526, November.
    • Handle: RePEc:spr:joptap:v:167:y:2015:i:2:d:10.1007_s10957-015-0763-3
    DOI: 10.1007/s10957-015-0763-3
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    References listed on IDEAS

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    1. X. M. Yang & D. Li & S. Y. Wang, 2001. "Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 413-427, August.
    2. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    3. César Gutiérrez & Rubén López & Vicente Novo, 2014. "Existence and Boundedness of Solutions in Infinite-Dimensional Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 515-547, August.
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