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Exposed extreme rays of the SONC cone

Mareike Dressler, Hongzhi Liao, Vera Roshchina

TL;DR

The paper addresses the structural understanding of the SONC cone by fully characterizing its exposed extreme rays on a finite ground set. It develops a purely combinatorial criterion based on the existence and nature of circuits, and delivers constructive exposing functionals via a graded partition of the ground set. The main contributions are a complete exposition of which extreme rays are exposed or not, along with explicit normals that certify exposure, enabling algorithmic detection. This advances both the theoretical geometry of SONC cones and practical SONC-based certification in polynomial optimization.

Abstract

We provide a complete and explicit characterization of the exposed extreme rays of the cone of sums of nonnegative circuit (SONC) polynomials. The criterion we derive is purely combinatorial and depends only on the existence of certain circuits within the ground set and on the nature of the corresponding extreme ray. Our constructive proofs also yield explicit exposing functionals, offering a basis for algorithmic detection of exposed rays in SONC-based optimization.

Exposed extreme rays of the SONC cone

TL;DR

The paper addresses the structural understanding of the SONC cone by fully characterizing its exposed extreme rays on a finite ground set. It develops a purely combinatorial criterion based on the existence and nature of circuits, and delivers constructive exposing functionals via a graded partition of the ground set. The main contributions are a complete exposition of which extreme rays are exposed or not, along with explicit normals that certify exposure, enabling algorithmic detection. This advances both the theoretical geometry of SONC cones and practical SONC-based certification in polynomial optimization.

Abstract

We provide a complete and explicit characterization of the exposed extreme rays of the cone of sums of nonnegative circuit (SONC) polynomials. The criterion we derive is purely combinatorial and depends only on the existence of certain circuits within the ground set and on the nature of the corresponding extreme ray. Our constructive proofs also yield explicit exposing functionals, offering a basis for algorithmic detection of exposed rays in SONC-based optimization.

Paper Structure

This paper contains 7 sections, 9 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonnegative circuit polynomials
  4. The SONC cone and its extreme rays
  5. Unexposed rays
  6. Exposed extreme rays
  7. Conclusion and open questions

Key Result

Theorem 1.1

An extreme ray $r$ of the SONC cone on a finite ground set $\mathcal{A}\subseteq\mathbb{N}^{n}$ is not exposed if and only if $r=\mathbb{R}_+ \mathbf{x}^{\boldsymbol{\alpha}}$ for some ${\boldsymbol{\alpha}}\in \mathcal{A}$, and there exists a circuit $(S,{\boldsymbol{\beta}})$ on $\mathcal{A}$ such

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2: katthan2021unified, Corollary 4.6
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 7 more