Vector optimization problems with quasiconvex constraints

Abstract

Let X be a real linear space, ${X_0 \subset X}$ a convex set, Y and Z topological real linear spaces. The constrained optimization problem min _C f(x), ${g(x) \in -K}$ is considered, where f : X ₀→ Y and g : X ₀→ Z are given (nonsmooth) functions, and ${C \subset Y}$ and ${K \subset Z}$ are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones.