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A unified concept of approximate and quasi efficient solutions and associated subdifferentials in multiobjective optimization

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Abstract

In this paper, we introduce some new notions of quasi efficiency and quasi proper efficiency for multiobjective optimization problems that reduce to the most important concepts of approximate and quasi efficient solutions given up to now. We establish main properties and provide characterizations for these solutions by linear and nonlinear scalarizations. With the help of quasi efficient solutions, a generalized subdifferential of a vector mapping is introduced, which generates a number of approximate subdifferentials frequently used in optimization in a unifying way. The generalized subdifferential is related to the classical subdifferential of real functions by the method of scalarization. An application of generalized subdifferential to express optimality conditions for quasi efficient solutions is also given.

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Acknowledgements

This work, for the first, second and fourth authors, was partially supported by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under project PGC2018-096899-B-I00 (MCIU/AEI/FEDER, UE), and also by ETSI Industriales, Universidad Nacional de Educación a Distancia (Spain) under Grant 2020-Mat09. The third author thanks his coauthors for invitation to joint the work and for their hospitality during his visit to the Department of Applied Mathematics at UNED, Madrid. The authors are very grateful to the referees for their useful suggestions and comments.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia (UNED), Calle Juan del Rosal, 12, 28040, Madrid, Spain

    L. Huerga, B. Jiménez & V. Novo

  2. Parametric MultiObjective Optimization Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    D. T. Luc

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    D. T. Luc

Authors
  1. L. Huerga
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  2. B. Jiménez
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  3. D. T. Luc
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  4. V. Novo
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Correspondence to V. Novo.

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Dedicated to Marco A. López on the occasion of his 70th birthday.

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Huerga, L., Jiménez, B., Luc, D.T. et al. A unified concept of approximate and quasi efficient solutions and associated subdifferentials in multiobjective optimization. Math. Program. 189, 379–407 (2021). https://doi.org/10.1007/s10107-020-01597-9

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  • DOI: https://doi.org/10.1007/s10107-020-01597-9

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