Nonlinear Cone Separation Theorems in Real Topological Linear Spaces

TL;DR

This paper develops new nonlinear separation theorems for cones in real topological linear spaces, extending classical convex separation results to non-convex cones using dual cones and separating functions.

Contribution

It introduces novel nonlinear cone separation theorems that generalize classical convex separation results to non-convex cones in topological linear spaces.

Findings

Derived new separation theorems for non-convex cones

Extended classical convex separation results to nonlinear settings

Utilized dual cones and separating functions in proofs

Abstract

The separation of two sets (or more specific of two cones) plays an important role in different fields of mathematics such as variational analysis, convex analysis, convex geometry, optimization. In the paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (topological) linear spaces. Basically, we follow the separation approach by Kasimbeyli (2010, SIAM J. Optim. 20) based on augmented dual cones and Bishop-Phelps type (normlinear) separating functions. Classical separation theorems for convex sets are the key tool for proving our main nonlinear cone separation theorems.