Verifying Global Optimality of Candidate Solutions to Polynomial Optimization Problems using a Determinant Relaxation Hierarchy

TL;DR

This paper introduces a determinant relaxation hierarchy method to efficiently verify the global optimality of solutions in polynomial optimization problems, including applications in power systems.

Contribution

It presents a novel, computationally tractable linear programming approach based on determinant relaxations within the Lasserre hierarchy for global optimality certification.

Findings

Effective verification of global optimality demonstrated on power flow problems

Determinant relaxation hierarchy simplifies the certification process

Method is computationally feasible for large polynomial problems

Abstract

We propose a method for verifying that a given feasible point for a polynomial optimization problem is globally optimal. The approach relies on the Lasserre hierarchy and the result of Lasserre regarding the importance of the convexity of the feasible set as opposed to that of the individual constraints. By focusing solely on certifying global optimality and relaxing the Lasserre hierarchy using necessary conditions for positive semidefiniteness based on matrix determinants, the proposed method is implementable as a computationally tractable linear program. We demonstrate this method via application to several instances of polynomial optimization, including the optimal power flow problem used to operate electric power systems.