Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to address this issue, Burer and Monteiro (Math Program 95(2):329–357, 2003) proposed to reduce the dimension of the problem by appealing to a low-rank factorization and solve the subsequent non-convex problem instead. In this paper, we present coordinate ascent based methods to solve this non-convex problem with provable convergence guarantees. More specifically, we prove that the block-coordinate maximization algorithm applied to the non-convex Burer–Monteiro method globally converges to a first-order stationary point with a sublinear rate without any assumptions on the problem. We further show that this algorithm converges linearly around a local maximum provided that the objective function exhibits quadratic decay. We establish that this condition generically holds when the rank of the factorization is sufficiently large. Furthermore, incorporating Lanczos method to the block-coordinate maximization, we propose an algorithm that is guaranteed to return a solution that provides \(1-{\mathcal {O}}\left( 1/r\right) \) approximation to the original SDP without any assumptions, where r is the rank of the factorization. This approximation ratio is known to be optimal (up to constants) under the unique games conjecture, and we can explicitly quantify the number of iterations to obtain such a solution.

具有对角约束的半定规划 (SDP) 出现在许多优化问题中，例如 Max-Cut、社区检测和组同步。尽管 SDP 可以在多项式时间内以任意精度求解，但通用凸求解器不能很好地适应问题的维度。为了解决这个问题，Burer 和 Monteiro (Math Program 95(2):329–357, 2003) 提出通过诉诸低秩分解来降低问题的维度，代之以解决随后的非凸问题。在本文中，我们提出了基于坐标上升的方法来解决这个具有可证明收敛保证的非凸问题。进一步来说，我们证明了应用于非凸 Burer-Monteiro 方法的块坐标最大化算法全局收敛到具有次线性速率的一阶驻点，而无需对该问题进行任何假设。我们进一步表明，如果目标函数呈现二次衰减，则该算法围绕局部最大值线性收敛。我们确定当分解的秩足够大时，这个条件一般成立。此外，结合 Lanczos 方法到块坐标最大化，我们提出了一种算法，保证返回一个解决方案，提供 我们确定当分解的秩足够大时，这个条件一般成立。此外，结合 Lanczos 方法到块坐标最大化，我们提出了一种算法，保证返回一个解决方案，提供 我们确定当分解的秩足够大时，这个条件一般成立。此外，结合 Lanczos 方法到块坐标最大化，我们提出了一种算法，保证返回一个解决方案，提供\(1-{\mathcal {O}}\left( 1/r\right) \)对原始 SDP 的近似，没有任何假设，其中r是因式分解的秩。在唯一博弈猜想下，这个近似比率已知是最优的（最多为常数），我们可以明确地量化迭代次数以获得这样的解决方案。