Second order necessary conditions in set constrained differentiable vector optimization

Abstract

We state second order necessary optimality conditions for a vector optimization problem with an arbitrary feasible set and an order in the final space given by a pointed convex cone with nonempty interior. We establish, in finite-dimensional spaces, second order optimality conditions in dual form by means of Lagrange multipliers rules when the feasible set is defined by a function constrained to a set with convex tangent cone. To pass from general conditions to Lagrange multipliers rules, a generalized Motzkin alternative theorem is provided. All the involved functions are assumed to be twice Fréchet differentiable.