Properties of the nonlinear scalar functional and its applications to vector optimization problems

Abstract

The nonconvex separation theorem for the Hiriart-Urruty nonlinear scalar functional (Hiriart-Urruty in Math Oper Res 4:79–97, 1979) and its sublinearity and monotonicity with respect to convex cone have played an important role in the research of vector optimization problems. By means of these special properties, this functional also can be used as a coherent measure of an investment. This work is devoted to investigating the properties of this special scalarizing function with respect to a co-radiant set, which is more general than a cone and is also a main tool in the study of approximate solutions in vector optimization problems. We first show that although this scalar functional with respect to the co-radiant set is not necessarily positively homogeneous, it satisfies some special properties, which imply it is a co-radiant function and also has subadditivity and monotonicity under the convexity assumption. Based on the subadditivity property, we calculate the classical (Fenchel) subdifferentials for the scalar functional. Finally, as the applications of our results, we give the nonlinear scalar characterizations for the approximate solutions of vector optimization problems, and establish the Lagrange multiplier rules in term of the Mordukhovich subdifferential in Asplund spaces.