Low-rank extragradient methods for scalable semidefinite optimization

We consider a class of important semidefinite optimization problems that involve a convex smooth or nonsmooth objective function and linear constraints. Focusing on high-dimensional settings with a low-rank solution that also satisfies a low-rank complementarity condition, we prove that the well-known Extragradient method, when initialized with a “warm-start”, converges with its standard convergence rate guarantees, using only efficient low-rank singular value decompositions to project onto the positive semidefinite cone. Supporting numerical evidence with a dataset of Max-Cut instances is provided.