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Vector critical points and efficiency in vector optimization with Lipschitz functions

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Abstract

In this work, we establish some relations between several notions of vector critical points and efficient, weak efficient and ideal efficient solutions of a vector optimization problem with a locally Lipschitz objective function. These relations are stated under pseudoinvexity hypotheses and via the generalized Jacobian. We provide a characterization of pseudoinvexity (resp. strong pseudoinvexity) through the property that every vector critical point is a weak efficient (resp. efficient) solution. We also obtain some properties of invex functions in connection with linear scalarizations. Several examples illustrating our results are also provided.

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Acknowledgments

The authors are grateful to the anonymous referees and the Associate Editor for their helpful comments and suggestions. This research was partially supported for the first three authors by the Ministerio de Economía y Competitividad (Spain) under project MTM2012-30942 and for the fourth author by Ministerio de Ciencia e Innovación (Spain) under project MTM2010-15383.

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Authors and Affiliations

  1. E.T.S. de Ingenieros de Telecomunicación, Universidad de Valladolid, Paseo de Belén 15, 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, 28040, Madrid, Spain

    B. Jiménez & V. Novo

  3. Departamento de Estadística e I.O, Universidad de Cádiz, Campus de Jerez de la Frontera, Avda. de la Universidad s/n, 11405, Jerez de la Frontera, Cádiz, Spain

    G. Ruiz-Garzón

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  1. C. Gutiérrez
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  2. B. Jiménez
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  3. V. Novo
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  4. G. Ruiz-Garzón
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Correspondence to V. Novo.

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Gutiérrez, C., Jiménez, B., Novo, V. et al. Vector critical points and efficiency in vector optimization with Lipschitz functions. Optim Lett 10, 47–62 (2016). https://doi.org/10.1007/s11590-015-0850-2

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  • DOI: https://doi.org/10.1007/s11590-015-0850-2

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