A Nonlinear Cone Separation Theorem and Scalarization in Nonconvex Vector Optimization

Abstract

In this paper, a special separation property for two closed cones in Banach spaces is proposed, and a nonlinear separation theorem for the cones possessing this property is proved. By extending a usual definition of dual cones, an augmented dual of a cone is introduced. A special class of monotonically increasing sublinear functions is defined by using the elements of the augmented dual cone. Any closed cone possessing the separation property with its $\varepsilon$-conic neighborhood is shown to be approximated arbitrarily closely by a zero sublevel set of some function from this class. As an application, a simple and efficient scalarization technique for nonconvex vector optimization problems is suggested, and it is shown that any properly minimal point of a set in a Banach space can be calculated by minimizing a certain sublinear functional.