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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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