📝 SummarXiv

Abstract

In the solving of large-scale semidefinite programs (SDPs), the symmetric Burer-Monteiro factorization (BMF) offers an economical low-rank solution of the form $XX^\top$. However, BMF turns a convex SDP into a more difficult nonconvex optimization problem in some cases, which limits the use of off-the-shelf convex optimization algorithms. To alleviate this problem, we convert symmetric BMF to its asymmetric counterpart involving $XY^\top$, and use a penalty with parameter $\gamma$ to encourage $X$ and $Y$ to be close. We show that the resultant asymmetric BMF is multi-convex, and thus convex optimization can again be used to solve the subproblems involving $X$ and $Y$ in an alternating manner. More importantly, to ensure that $X=Y$ on convergence, we derive theoretically sound conditions for exact $\gamma$ that are independent of both the application problem and underlying algorithm. Experiments demonstrate that the proposed method is more advantageous over existing related approaches.