Alternative Theorems and Necessary Optimality Conditions for Directionally Differentiable Multiobjective Programs

We study, in a unified way, some alternative theorems that involve linear and sublinear functions between finite dimensional spaces and a convex set, and we propose several generalizations of them. These theorems are applied to obtain, under different constraint qualifications, several necessary conditions for a point to be Pareto optimum, both Fritz John and Kuhn-Tucker type, in multiobjective programming problems which are defined by directionally differentiable functions and which include three types of constraints: inequality, equality and set constraints. In particular, these necessary conditions are applicable to convex programs and to differentiable programs.