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On gap functions for nonsmooth multiobjective optimization problems

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Abstract

A set-valued gap function, \(\phi \), existing in the literature for smooth and nonsmooth multiobjective optimization problems is dealt with. It is known that \(0\in \phi (x^*)\) is a sufficient condition for efficiency of a feasible solution \(x^*\), while the converse does not hold. In the current work, the converse of this assertion is proved for properly efficient solutions. Afterwards, to avoid the complexities of set-valued maps some new single-valued gap functions, for nonsmooth multiobjective optimization problems with locally Lipschitz data are introduced. Important properties of the new gap functions are established.

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Acknowledgements

The authors would like to express their gratitude to the anonymous referees for helpful comments on the first version of the paper. The research of the last author was in part supported by a grant from IPM (No. 95260124).

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Authors and Affiliations

  1. Department of Economics, University of Messina, via dei Verdi, 75, Messina, Italy

    Giuseppe Caristi

  2. Department of Mathematics, Payame Noor University (PNU), Tehran, Iran

    Nader Kanzi

  3. School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran

    Majid Soleimani-damaneh

  4. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

    Majid Soleimani-damaneh

Authors
  1. Giuseppe Caristi
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  2. Nader Kanzi
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  3. Majid Soleimani-damaneh
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Correspondence to Majid Soleimani-damaneh.

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Caristi, G., Kanzi, N. & Soleimani-damaneh, M. On gap functions for nonsmooth multiobjective optimization problems. Optim Lett 12, 273–286 (2018). https://doi.org/10.1007/s11590-017-1110-4

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  • DOI: https://doi.org/10.1007/s11590-017-1110-4

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