Nonlinear Strict Cone Separation Theorems in Real Normed Spaces

TL;DR

This paper introduces new nonlinear strict cone separation theorems in real normed spaces, extending previous results by weakening conditions and utilizing augmented dual cones and separating functions.

Contribution

It generalizes existing separation theorems by relaxing convexity and closedness assumptions using a novel characterization of the algebraic interior of augmented dual cones.

Findings

Established new separation theorems under weaker conditions.

Linked the new results to previous theorems by Kasimbeyli and others.

Provided a characterization of the algebraic interior of augmented dual cones.

Abstract

In this paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (reflexive) normed spaces. In essence, we follow the nonlinear and nonsymmetric separation approach developed by Kasimbeyli (2010, SIAM J. Optim. 20), which is based on augmented dual cones and Bishop-Phelps type (normlinear) separating functions. Compared to Kasimbeyli's separation theorem, we formulate our theorems for the separation of two cones under weaker conditions (concerning convexity and closedness requirements) with respect to the involved cones. By a new characterization of the algebraic interior of augmented dual cones in real normed spaces, we are able to establish relationships between our cone separation results and the results derived by Kasimbeyli (2010, SIAM J. Optim. 20) and by Garcia-Castano, Melguizo-Padial and Parzanese (2023, Math. Meth. Oper. Res. 97).