  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.
