Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization

Abstract

This paper investigates vector optimization problems with objective and the constraints are multifunctions. By using a special scalarization function introduced in optimization by Hiriart-Urruty, we establish optimality conditions in terms of Lagrange–Fritz–John and Lagrange–Kuhn–Tucker multipliers. When all the data of the problem are subconvexlike we derive the results by Li, and hence those of Lin and Corley. We also show how the generalized Moreau–Rockafellar type theorem to multifunctions obtained recently by Lin can be derived from the well-known results in scalar optimization. In the last, vector optimization problem in which objective and the constraints are defined by multifunctions and depends on a parameter u, and the resulting value multifunction M( u) are considered. With the help of the generalized Moreau–Rockafellar type theorem we establish the weak subdifferential of M in terms of the weak subdifferential of objective and constraint multifunctions.