Step-affine functions, halfspaces, and separation of convex sets with applications to convex optimization problems
The paper presents analytical criteria for the disjointness of convex subsets of a real vector space. The criteria are based on the separation of convex sets by step-linear and step-affine functions. These criteria are essentially analytic versions of the Kakutani-Tukey criterion on the separation of convex sets by halfspaces and these are important due to their applications to irregular convex optimization problems. As applications of these results, the author derives a minimality criterion for solutions of convex vector optimization problems considered in real vector spaces without topology and an optimality criterion for admissible points in classical convex programming problems not satisfying the Slater regularity condition. Some open questions related to the separation of convex sets by halfspaces are also given.
scientific article; zbMATH DE number 7377484
scientific article