Optimality and duality in nonsmooth semi-infinite optimization, using a weak constraint qualification

Variational analysis, a subject that has been vigorously developing for the past 40 years, has proven itself to be extremely effective at describing nonsmooth phenomenon. The Clarke subdifferential (or generalized gradient) and the limiting subdifferential of a function are the earliest and most widely used constructions of the subject. A key distinction between these two notions is that, in contrast to the limiting subdifferential, the Clarke subdifferential is always convex. From a computational point of view, convexity of the Clarke subdifferential is a great virtue. We consider a nonsmooth multiobjective semi-infinite programming problem with a feasible set defined by inequality constraints. First we introduce the weak Slater constraint qualification, and derive the Karush-Kuhn-Tucker types necessary and sufficient conditions for (weakly, properly) efficient solution of the considered problem. Then, we introduce two duals of Mond-Weir type for the problem, a present (weak and strong) duality results for them. All results are given in term of Clarke subdifferential.