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Nonlinear Strict Cone Separation Theorems in Real Normed Spaces

Christian Günther, Bahareh Khazayel, Christiane Tammer

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).

Nonlinear Strict Cone Separation Theorems in Real Normed Spaces

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).
Paper Structure (9 sections, 14 theorems, 69 equations, 1 figure)

This paper contains 9 sections, 14 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Fundamentals of Real Normed Spaces
  4. Convex Sets, Cones and their Bases
  5. Classical Separation Theorems for Convex Sets / Cones
  6. Properties of Augmented Dual Cones
  7. Nonlinear Separation Approach based on Bishop-Phelps Cones
  8. Strict Cone Separation Theorems in Real (Reflexive) Normed Spaces
  9. Conclusions

Key Result

Theorem 2.1

Assume that $E$ is a real normed space, $K \subseteq E$ is a nontrivial cone, and $B_{K}$ is the norm-base of $K$. Then, the following assertions are valid:

Figures (1)

  • Figure 1: The nontrivial, closed, pointed, solid, convex cone $-K = \mathbb{R}^2_+$ and the nontrivial, closed, solid, nonconvex cone $A$ in the real normed space $(\mathbb{R}^2, ||\cdot||_2)$ are strictly separated by $C_{x^*, \alpha}^{\leq}$. In particular, the condition \ref{['eq:-clSAcapclSK=empty2']} is valid.

Theorems & Definitions (61)

  • Example 1.1
  • Remark 1.1
  • Remark 2.1: Strong and weak topologies
  • Remark 2.2: Strong and weak topologies
  • Definition 2.1
  • Remark 2.3
  • Definition 2.2
  • Remark 2.4
  • Definition 2.3
  • Remark 2.5
  • ...and 51 more