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Sparse Copositive Polynomial Optimization

Suhan Zhong, Jinling Zhou, Jiawang Nie, Xindong Tang

Abstract

This paper studies the copositive optimization problem whose objective is a sparse polynomial, with linear constraints over the nonnegative orthant. We propose sparse Moment-SOS relaxations to solve it. Necessary and sufficient conditions are shown for these relaxations to be tight. In particular, we prove they are tight under the cop-SOS convexity assumption. Compared to the traditional dense ones, the sparse Moment-SOS relaxations are more computationally efficient. Numerical experiments are given to show the efficiency.

Sparse Copositive Polynomial Optimization

Abstract

This paper studies the copositive optimization problem whose objective is a sparse polynomial, with linear constraints over the nonnegative orthant. We propose sparse Moment-SOS relaxations to solve it. Necessary and sufficient conditions are shown for these relaxations to be tight. In particular, we prove they are tight under the cop-SOS convexity assumption. Compared to the traditional dense ones, the sparse Moment-SOS relaxations are more computationally efficient. Numerical experiments are given to show the efficiency.

Paper Structure

This paper contains 8 sections, 6 theorems, 120 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard Moment-SOS relaxations
  4. Sparse polynomials and moments
  5. The sparse Moment-SOS relaxations
  6. The cop-SOS convexity
  7. Numerical experiments
  8. Conclusions

Key Result

Theorem 3.1

For the $k$th order sparse SOS relaxation eq:spcopsos, we have: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (22)

  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Example 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Example 3.6
  • Theorem 4.1
  • proof
  • ...and 12 more