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On the Existence of Weak Efficient Solutions of Nonconvex Vector Optimization Problems

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Abstract

We study vector optimization problems with solid non-polyhedral convex ordering cones, without assuming any convexity or quasiconvexity assumption. We state a Weierstrass-type theorem and existence results for weak efficient solutions for coercive and noncoercive problems. Our approach is based on a new coercivity notion for vector-valued functions, two realizations of the Gerstewitz scalarization function, asymptotic analysis and a regularization of the objective function. We define new boundedness and lower semicontinuity properties for vector-valued functions and study their properties. These new tools rely heavily on the solidness of the ordering cone through the notion of colevel and level sets. As a consequence of this approach, we improve various existence results from the literature, since weaker assumptions are required.

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Acknowledgements

The authors are very grateful to the anonymous referees for their helpful comments and suggestions. This work was carried out mainly in July 2017 and July 2019, during two research stays of the first author in the Departamento de Matemática at Universidad de Tarapacá, Arica, Chile. He wish to thank all the staff for its warm hospitality. This research was partially supported by Ministerio de Economía y Competitividad (Spain) under Project MTM2015-68103-P (MINECO/FEDER), by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER, UE) under Project PGC2018-096899-B-I00 and for the second author by ANID (Chile) under Project Fondecyt 1181368.

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Authors and Affiliations

  1. University of Valladolid, Valladolid, Spain

    César Gutiérrez

  2. Universidad de Tarapacá, Arica, Chile

    Rubén López

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  1. César Gutiérrez
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  2. Rubén López
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Correspondence to César Gutiérrez.

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Communicated by Radu Ioan Boţ

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Gutiérrez, C., López, R. On the Existence of Weak Efficient Solutions of Nonconvex Vector Optimization Problems. J Optim Theory Appl 185, 880–902 (2020). https://doi.org/10.1007/s10957-020-01667-0

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  • DOI: https://doi.org/10.1007/s10957-020-01667-0

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