🤖 AI Summary
Standard semidefinite programming (SDP) solvers suffer from poor scalability, while the conventional Burer–Monteiro factorization (BMF) introduces nonconvexity, hindering reliable optimization.
Method: We propose a biconvex optimization framework based on the bilinear decomposition $Z = XY^ op$, augmented with a structural penalty term $|X - Y|_F^2$ to ensure tractable optimization.
Contribution/Results: This work is the first to explicitly formulate SDPs as biconvex problems. We derive a theoretically grounded upper bound on the penalty parameter $gamma$, guaranteeing equivalence to low-rank BMF solutions at stationarity. Our approach establishes a novel biconvex surrogate paradigm for SDPs. Integrated with alternating minimization, it achieves state-of-the-art performance on matrix completion and Max-Cut—two canonical SDP tasks—while significantly improving both efficiency and accuracy for large-scale instances.
📝 Abstract
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 $XX^ op$). 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 $XY^ op$), 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.