Deterministic guarantees for Burer-Monteiro factorizations of smooth semidefinite programs

We consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix X of size n. Following the Burer--Monteiro approach, we optimize a factor Y of size n × p instead, such that X = YYT. This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if p is small, but results in a non-convex optimization problem with a quadratic cost function and quadratic equality constraints in Y. In this paper, we show that if the set of constraints on Y regularly defines a smooth manifold, then, despite non-convexity, first- and second-order necessary optimality conditions are also sufficient, provided p is large enough. For smaller values of p, we show a similar result holds for almost all (linear) cost functions. Under those conditions, a global optimum Y maps to a global optimum X = YYT of the SDP. We deduce old and new consequences for SDP relaxations of the generalized eigenvector problem, the trust-region subproblem and quadratic optimization over several spheres, as well as for the Max-Cut and Orthogonal-Cut SDPs which are common relaxations in stochastic block modeling and synchronization of rotations.