Nonsmooth Vector Optimization Problems

Abstract

This chapter deals with the vector optimization problems having set-valued objectives. We develop the theory of contingent derivatives for set-valued maps and then study local optimality conditions for these problems via contingent derivatives. The concept of contingent cones was introduced by Bouligand as early as in the thirties. However, it has got a wide application in nonsmooth analysis very recently, due to the work of Aubin (1981) and to the development of other concepts of tangent cones in the study of nondifferentiable functions (Aubin and Ekeland -1984, Prankowska -1985, Penot -1984, Ward and Borwein-1987). Contingent cones suffer from an undesirable defect of being nonconvex, hence many powerful techniques of convex analysis cannot be exploited when using these cones to define derivatives of functions. Nevertheless, since containing almost every existing tangent cone, they carry in themselves rich information about the local behavior of sets and they always exist regardless of the structure of the sets. Moreover, these cones enjoy enough properties to make a decent calculus and they axe well suited to define derivatives of set-valued functions. This is why we choose contingent derivatives in order to produce optimality conditions for vector problems with set-valued data.