On gap functions for nonsmooth multiobjective optimization problems

A set-valued gap function, $$\phi $$ϕ, existing in the literature for smooth and nonsmooth multiobjective optimization problems is dealt with. It is known that $$0\in \phi (x^*)$$0∈ϕ(x∗) is a sufficient condition for efficiency of a feasible solution $$x^*$$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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