非凸半定优化的一种有效的乘子交替方向法
An efficient approach for nonconvex semidefinite optimization via customized alternating direction method of multipliers
摘要：
我们研究了一类一般的组合图问题，包括MAX-CUT和社区检测，重新制定为二次目标的非凸约束，并通过交替方向乘法（ADMM）解决。我们提出两个重新表述：一个使用向量变量和二进制约束，和其他进一步重新制定Burer-Monteiro形式为更简单的子问题。尽管非凸约束，我们证明了ADMM迭代收敛到一个稳定点，在这两个配方，在温和的假设。此外，最近的工作表明，在后一种形式中，当矩阵因子足够宽时，具有高概率的局部最优也是全局最优。为了证明我们的算法的可扩展性，我们包括最大切割，社区检测，图像分割基准和模拟的例子。
英文摘要：
We investigate a class of general combinatorial graph problems, including MAX-CUT and community detection, reformulated as quadratic objectives over nonconvex constraints and solved via the alternating direction method of multipliers (ADMM). We propose two reformulations: one using vector variables and a binary constraint, and the other further reformulating the Burer-Monteiro form for simpler subproblems. Despite the nonconvex constraint, we prove the ADMM iterates converge to a stationary point in both formulations, under mild assumptions. Additionally, recent work suggests that in this latter form, when the matrix factors are wide enough, local optimum with high probability is also the global optimum. To demonstrate the scalability of our algorithm, we include results for MAX-CUT, community detection, and image segmentation benchmark and simulated examples.
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