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

Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer–Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the penalty formulation of appropriately rank-constrained SDPs as long as the number of constraints scales sub-quadratically with the desired rank of the optimal solution. Our result is based on a simple penalty function formulation of the rank-constrained SDP along with a smoothed analysis to avoid worst-case cost matrices. We particularize our results to two applications, namely, Max-Cut and matrix completion.