On gap functions for nonsmooth multiobjective optimization problems (Q1744623)

It is known that, if at some feasible solution of a differentiable multiobjective problem the set-valued gap function introduced by \textit{G. Y. Chen} et al. [Eur. J. Oper. Res. 111, No. 1, 142--151 (1998; Zbl 0944.90079)] contains zero, then that solution is efficient. In the present work, the authors prove that the converse statement is true if the solution is proper, and give a nonsmooth version of gap functions for convex problems. They also introduce a single-valued gap function and show that a feasible solution of a quasiconvex problem is efficient if and only if the gap function takes the value zero at that solution, provided that the so-called nonvanishing constraint qualification (NCQ) is satisfied. Note that the condition (NCQ) is not applicable to unconstrained single-criterion problems.