Completely positive and completely positive semidefinite tensor relaxations for polynomial optimization

TL;DR

This paper explores tensor-based relaxations for polynomial optimization problems, demonstrating their potential to provide tighter bounds than traditional matrix-based methods through theoretical analysis and numerical experiments.

Contribution

It introduces CP and CPSD tensor relaxations for general POPs and compares their effectiveness with existing Lagrangian relaxations, extending quadratic POP results to higher degrees.

Findings

Tensor relaxations can yield tighter bounds than quadratic reformulations.

CP and CPSD tensor relaxations extend known quadratic results to general POPs.

Numerical experiments show improved bounds on small-scale problems.

Abstract

Completely positive (CP) tensors, which correspond to a generalization of CP matrices, allow to reformulate or approximate a general polynomial optimization problem (POP) with a conic optimization problem over the cone of CP tensors. Similarly, completely positive semidefinite (CPSD) tensors, which correspond to a generalization of positive semidefinite (PSD) matrices, can be used to approximate general POPs with a conic optimization problem over the cone of CPSD tensors. In this paper, we study CP and CPSD tensor relaxations for general POPs and compare them with the bounds obtained via a Lagrangian relaxation of the POPs. This shows that existing results in this direction for quadratic POPs extend to general POPs. Also, we provide some tractable approximation strategies for CP and CPSD tensor relaxations. These approximation strategies show that, with a similar computational effort, bounds obtained from them for general POPs can be tighter than bounds for these problems obtained by reformulating the POP as a quadratic POP, which subsequently can be approximated using CP and PSD matrices. To illustrate our results, we numerically compare the bounds obtained from these relaxation approaches on small scale fourth-order degree POPs.