Weak Conjugate Duality for Nonconvex Vector Optimization

Abstract

In this work, weak conjugate map, weak biconjugate map and weak subdifferential of a set-valued map are defined by using notions of supremum/infimum of a set and vectorial norm, and relationships among these notions are examined. Furthermore, necessary and sufficient conditions for weakly subdifferentiability of a set-valued map are given. It is proved that under some assumptions Lipschitz set-valued maps are weakly subdifferentiable. By using these notions a dual of unconstrained nonconvex vector optimization problems is constructed, and weak duality theorem is presented. Stability of primal problem is defined and it is proved that the stability of primal problem implies the strong duality. Furthermore, some stability conditions are presented. By using a special perturbation function weak Fenchel dual problem of constrained vector optimization problem is constructed and at the end, an example of a nonconvex constrained vector optimization problem which can not be solved by using Lagrange dual problem [25] but can be solved by using weak Fenchel dual problem is given.