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A finite-termination algorithm for testing copositivity over the positive semidefinite cone

Lei Huang, Lingling Xie

TL;DR

The paper addresses the challenging problem of testing $\\mathcal{S}_+^n$-copositivity of homogeneous polynomials and extends to the product cone $\\mathcal{S}_+^n\\times \\mathbb{R}_+^m$. It develops a strengthened reformulation derived from first-order optimality conditions and deploys a matrix Moment-SOS relaxation hierarchy that is proven to terminate in finite steps. The resulting algorithms either certify copositivity or return a refuting point with a PSD matrix $X(u)$ and negative polynomial value, with analogous extensions to the product cone. Numerical experiments demonstrate the method’s effectiveness and finite termination in practice, suggesting a viable, certificate-providing approach for copositivity problems in polynomial optimization contexts.

Abstract

This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of semidefinite programs. Notably, it always terminates in finitely many iterations. If a homogeneous polynomial is copositive over the positive semidefinite cone, the algorithm provides a certificate; otherwise, it returns a vector that refutes copositivity. Building on a similar idea, we further propose an algorithm to test copositivity over the direct product of the positive semidefinite cone and the nonnegative orthant. Preliminary numerical experiments demonstrate the effectiveness of the proposed methods.

A finite-termination algorithm for testing copositivity over the positive semidefinite cone

TL;DR

The paper addresses the challenging problem of testing -copositivity of homogeneous polynomials and extends to the product cone . It develops a strengthened reformulation derived from first-order optimality conditions and deploys a matrix Moment-SOS relaxation hierarchy that is proven to terminate in finite steps. The resulting algorithms either certify copositivity or return a refuting point with a PSD matrix and negative polynomial value, with analogous extensions to the product cone. Numerical experiments demonstrate the method’s effectiveness and finite termination in practice, suggesting a viable, certificate-providing approach for copositivity problems in polynomial optimization contexts.

Abstract

This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of semidefinite programs. Notably, it always terminates in finitely many iterations. If a homogeneous polynomial is copositive over the positive semidefinite cone, the algorithm provides a certificate; otherwise, it returns a vector that refutes copositivity. Building on a similar idea, we further propose an algorithm to test copositivity over the direct product of the positive semidefinite cone and the nonnegative orthant. Preliminary numerical experiments demonstrate the effectiveness of the proposed methods.
Paper Structure (14 sections, 10 theorems, 161 equations, 5 tables)

This paper contains 14 sections, 10 theorems, 161 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimality conditions for nonlinear semidefinite optimization
  4. Some basics in real algebraic geometry
  5. The matrix Moment-SOS relaxations
  6. A strengthened reformulation and Moment-SOS relaxations
  7. A strengthened reformulation
  8. The matrix Moment--SOS hierarchy of (\ref{['copo:equ']})
  9. An algorithm for testing $\mathcal{S}^n_{+}$-copositivity
  10. An algorithm for testing $\mathcal{S}_+^n\times \mathbb{R}_+^m$-copositivity
  11. Numerical examples
  12. The case that $K=\mathcal{S}_+^n$
  13. The case that $K=\mathcal{S}_+^n\times \mathbb{R}_+^m$
  14. Conclusions and discussions

Key Result

Proposition 2.1

Suppose $u$ is a feasible point of (nsdp) and $\hbox{rank}~ G_t(u)=r_t$ for $t=1,\dots,s$. Let $\{q_1^{(t)},\dots,q_{m_t-r_t}^{(t)}\}$ be an arbitrary basis for the kernel of $G_t(u)$. Then, the NDC (CQ) holds at $u$ if and only if the following vectors are linearly independent:

Theorems & Definitions (24)

  • Proposition 2.1: shunisunde
  • Theorem 2.2: shapsunde
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 4.2
  • Theorem 4.3
  • ...and 14 more