A biconvex optimization for solving semidefinite programs via bilinear factorization

Many problems in machine learning can be reduced to learning a low-rank positive semidefinite matrix (denoted as $Z$), which encounters semidefinite program (SDP). Existing SDP solvers by classical convex optimization are expensive to solve large-scale problems. Employing the low rank of solution, Burer-Monteiro's method reformulated SDP as a nonconvex problem via the $quadratic$ factorization ($Z$ as $XX^\top$). However, this would lose the structure of problem in optimization. In this paper, we propose to convert SDP into a biconvex problem via the $bilinear$ factorization ($Z$ as $XY^\top$), and while adding the term $\frac{\gamma}{2}||X-Y||_F^2$ to penalize the difference of $X$ and $Y$. Thus, the biconvex structure (w.r.t. $X$ and $Y$) can be exploited naturally in optimization. As a theoretical result, we provide a bound to the penalty parameter $\gamma$ under the assumption of $L$-Lipschitz smoothness and $\sigma $-strongly biconvexity, such that, at stationary points, the proposed bilinear factorization is equivalent to Burer-Monteiro's factorization when the bound is arrived, that is $\gamma>\frac{1}{4}(L-\sigma)_+$. Our proposal opens up a new way to surrogate SDP by biconvex program. Experiments on two SDP-related applications demonstrate that the proposed method is effective as the state-of-the-art.