On the local stability of semidefinite relaxations

In this paper we consider a parametric family of polynomial optimization problems over algebraic sets. Although these problems are typically nonconvex, tractable convex relaxations via semidefinite programming (SDP) have been proposed. Often times in applications there is a natural value of the parameters for which the relaxation will solve the problem exactly. We study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood of the original one. It suffices to restrict to quadratically constrained quadratic programs. Our framework captures several estimation problems such as low rank approximation, camera triangulation, rotation synchronization, approximate matrix completion and approximate GCD. In these applications, a solution is easy under noiseless observations, and our results guarantee that the SDP relaxation will continue to solve the problem in the low noise regime.



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