Sparse Noncommutative Polynomial Optimization

TL;DR

This paper develops a hierarchy of semidefinite relaxations for optimizing polynomials in noncommuting variables, incorporating sparsity, and provides methods to compute bounds on eigenvalues and traces using operator algebra techniques.

Contribution

It introduces a converging hierarchy of relaxations for noncommutative polynomial optimization that accounts for sparsity, extending classical results to the noncommutative setting.

Findings

Provides a hierarchy of semidefinite relaxations for eigenvalue and trace optimization.

Utilizes GNS construction to extract optimizers under certain conditions.

Computes lower bounds on eigenvalues and traces of noncommutative polynomials.

Abstract

This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre [SIAM J. Optim. 17(3) (2006), pp. 822--843] and Waki et al. [SIAM J. Optim. 17(1) (2006), pp. 218--242]. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue and trace of noncommutative polynomials from the literature.