ε-Subdifferentials of set-valued maps and ε-weak Pareto optimality for multiobjective optimization

In this paper we consider vector optimization problems where objective and constraints are set-valued maps. Optimality conditions in terms of Lagrange-multipliers for an epsilon-weak Pareto minimal point are established in the general case and in the case with nearly subconvexlike data. A comparison with existing results is also given. Our method used a special scalarization function, introduced in optimization by Hiriart-Urruty. Necessary and sufficient conditions for the existence of an epsilon-weak Pareto minimal point are obtained. The relation between the set of all epsilon-weak Pareto minimal points and the set of all weak Pareto minimal points is established. The epsilon-subdifferential formula of the sum of two convex functions is also extended to set-valued maps via well known results of scalar optimization. This result is applied to obtain the Karush-Kuhn-Tucker necessary conditions, for epsilon-weak Pareto minimal points