Low-Rank Extragradient Methods for Scalable Semidefinite Optimization
Dan Garber. Atara Kaplan
We consider several classes of highly important semidefinite optimization
problems that involve both a convex objective function (smooth or nonsmooth)
and additional linear or nonlinear smooth and convex constraints, which are
ubiquitous in statistics, machine learning, combinatorial optimization, and
other domains. We focus on high-dimensional and plausible settings in which the
problem admits a low-rank solution which also satisfies a low-rank
complementarity condition. We provide several theoretical results proving that,
under these circumstances, the well-known Extragradient method, when
initialized in the proximity of an optimal primal-dual solution, converges to a
solution of the constrained optimization problem with its standard convergence
rates guarantees, using only low-rank singular value decompositions (SVD) to
project onto the positive semidefinite cone, as opposed to
computationally-prohibitive full-rank SVDs required in worst-case. Our approach
is supported by numerical experiments conducted with a dataset of Max-Cut
instances.