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Semidefinite programs (SDPs) can be solved in polynomial time by interiorpoint methods, but scalability can be an issue. To address this shortcoming,over a decade ago, Burer and Monteiro proposed to solve SDPs with few equalityconstraints via rank-restricted, non-convex surrogates. Remarkably, for someapplications, local optimization methods seem to converge to global optima ofthese non-convex surrogates reliably. Although some theory supports thisempirical success, a complete explanation of it remains an open question. Inthis paper, we consider a class of SDPs which includes applications such asmax-cut, community detection in the stochastic block model, robust PCA, phaseretrieval and synchronization of rotations. We show that the low-rankBurer--Monteiro formulation of SDPs in that class almost never has any spuriouslocal optima.

The non-convex Burer-Monteiro approach works on smooth semidefinite programs