Exploiting constant trace property in large-scale polynomial optimization

TL;DR

This paper introduces a reformulation of polynomial optimization problems with ball constraints into semidefinite programs with a constant trace property, enabling more efficient solutions via first-order methods, especially for large-scale problems.

Contribution

It proves the constant trace property for semidefinite relaxations of POPs with ball constraints and extends the framework to large-scale problems with various sparsity structures.

Findings

Efficient solution of large-scale POPs using CTP-exploiting methods.

Demonstrated scalability on randomly generated quadratically constrained quadratic programs.

Improved computational performance over traditional methods.

Abstract

We prove that every semidefinite moment relaxation of a polynomial optimization problem (POP) with a ball constraint can be reformulated as a semidefinite program involving a matrix with constant trace property (CTP). As a result such moment relaxations can be solved efficiently by first-order methods that exploit CTP, e.g., the conditional gradient-based augmented Lagrangian method. We also extend this CTP-exploiting framework to large-scale POPs with different sparsity structures. The efficiency and scalability of our framework are illustrated on second-order moment relaxations for various randomly generated quadratically constrained quadratic programs.