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Approximate solutions of vector optimization problems via improvement sets in real linear spaces

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Abstract

We deal with a constrained vector optimization problem between real linear spaces without assuming any topology and by considering an ordering defined through an improvement set E. We study E-optimal and weak E-optimal solutions and also proper E-optimal solutions in the senses of Benson and Henig. We relate these types of solutions and we characterize them through approximate solutions of scalar optimization problems via linear scalarizations and nearly E-subconvexlikeness assumptions. Moreover, in the particular case when the feasible set is defined by a cone-constraint, we obtain characterizations by means of Lagrange multiplier rules. The use of improvement sets allows us to unify and to extend several notions and results of the literature. Illustrative examples are also given.

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Acknowledgements

The authors are grateful to the anonymous referee and the Associated Editor for their useful suggestions and remarks.

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Authors and Affiliations

  1. IMUVA (Institute of Mathematics of University of Valladolid), Paseo de Belén 7, Campus Miguel Delibes, 47011, Valladolid, Spain

    C. Gutiérrez

  2. Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia (UNED), c/ Juan del Rosal 12, Ciudad Universitaria, 28040, Madrid, Spain

    L. Huerga, B. Jiménez & V. Novo

Authors
  1. C. Gutiérrez
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  2. L. Huerga
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  3. B. Jiménez
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  4. V. Novo
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Corresponding author

Correspondence to C. Gutiérrez.

Additional information

This research was partially supported by Ministerio de Economía y Competitividad (Spain) under Project MTM2015-68103-P (MINECO/FEDER).

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Gutiérrez, C., Huerga, L., Jiménez, B. et al. Approximate solutions of vector optimization problems via improvement sets in real linear spaces. J Glob Optim 70, 875–901 (2018). https://doi.org/10.1007/s10898-017-0593-y

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  • DOI: https://doi.org/10.1007/s10898-017-0593-y

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