Deterministic guarantees for Burer-Monteiro factorizations of smooth semidefinite programs.

*(English)*Zbl 1445.90074Summary: 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 \(n \times p\) instead, such that \(X = YY^T\). This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if \(p\) is small, but results in a nonconvex 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 nonconvexity, 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 \(X = YY^T\) 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.