The Constant Trace Property in Noncommutative Optimization

TL;DR

This paper introduces a constant trace property for semidefinite relaxations of noncommutative polynomial optimization problems, enabling more efficient solutions through first-order numerical methods.

Contribution

It establishes a novel constant trace formulation for semidefinite relaxations, facilitating improved computational efficiency in noncommutative optimization.

Findings

Semidefinite relaxations can be reformulated with a constant trace matrix variable.

First-order methods can exploit this property for faster solutions.

The approach enhances the computational tractability of noncommutative polynomial optimization.

Abstract

In this article, we show that each semidefinite relaxation of a ball-constrained noncommutative polynomial optimization problem can be cast as a semidefinite program with a constant trace matrix variable. We then demonstrate how this constant trace property can be exploited via first order numerical methods to solve efficiently the semidefinite relaxations of the noncommutative problem.