Surrogate duality for quasiconvex vector minimization

Abstract

It is well-known that duality theory is a fundamental tool in various areas of mathematics. There are great advantages to including or using the dual problem and duality statements. Especially, solving the dual problem can be done using other methods of analysis or numerical mathematics. We consider a primal vector optimization problem with an objective function acting between a linear topological space X and a linear topological space Y equipped with a pointed closed convex cone $$D\subset Y$$ D ⊂ Y with nonempty interior. The feasible set is supposed to be a closed convex cone. The aim of this paper is to construct a simple and easy-to-handle dual problem by exploiting the special structure of the primal problem and using a suitable nonlinear scalarization. The computation of the dual image set involves the minimization of a nonlinear scalarization of the primal vector-valued objective function subject to only one linear inequality constraint. We introduce a new concept of upper semicontinuity for a vector-valued function and prove (weak and strong) duality statements under the assumption that the vector-valued objective function is D -quasiconvex and D -upper semicontinuous. Furthermore, we study special cases.