Global optimization of low-rank polynomials

TL;DR

This paper introduces LRPOP, a hierarchy of semidefinite relaxations tailored for low-rank polynomial optimization problems, enabling the solution of large-scale problems previously intractable by existing methods.

Contribution

The paper presents LRPOP, a novel hierarchy of relaxations that leverages low-rank tensor decompositions to efficiently solve large polynomial optimization problems.

Findings

Can solve problems with thousands of variables and degrees

Improved numerical conditioning using Bernstein basis

Hierarchy converges to the global minimum from below

Abstract

This work considers polynomial optimization problems where the objective admits a low-rank canonical polyadic tensor decomposition. We introduce LRPOP (low-rank polynomial optimization), a new hierarchy of semidefinite programming relaxations for which the size of the semidefinite blocks is determined by the canonical polyadic rank rather than the number of variables. As a result, LRPOP can solve low-rank polynomial optimization problems that are far beyond the reach of existing sparse hierarchies. In particular, we solve problems with up to thousands of variables with total degree in the thousands. Numerical conditioning for problems of this size is improved by using the Bernstein basis. The LRPOP hierarchy converges from below to the global minimum of the polynomial under standard assumptions.