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Semidefinite programs (SDP) are important in learning and combinatorialoptimization with numerous applications. In pursuit of low-rank solutions andlow complexity algorithms, we consider the Burer--Monteiro factorizationapproach for solving SDPs. We show that all approximate local optima are globaloptima for the penalty formulation of appropriately rank-constrained SDPs aslong as the number of constraints scales sub-quadratically with the desiredrank of the optimal solution. Our result is based on a simple penalty functionformulation of the rank-constrained SDP along with a smoothed analysis to avoidworst-case cost matrices. We particularize our results to two applications,namely, Max-Cut and matrix completion.

Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form