On the local stability of semidefinite relaxations

TL;DR

This paper investigates the local stability of semidefinite relaxations for a broad class of quadratically constrained quadratic programs, providing conditions under which the relaxations remain exact near a nominal parameter value.

Contribution

It introduces a framework to analyze the stability of SDP relaxations for QCQPs, applicable to various statistical estimation problems and polynomial optimization.

Findings

Conditions for SDP relaxation stability near a nominal parameter

Quantitative bounds for relaxation exactness

Applicability to diverse statistical estimation problems

Abstract

We consider a parametric family of quadratically constrained quadratic programs (QCQP) and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems.