We present an online algorithm for time-varying semidefinite programs (TV-SDPs), based on the tracking of the solution trajectory of a low-rank matrix factorization, also known as the Burer-Monteiro factorization, in a path-following procedure. There, a predictor-corrector algorithm solves a sequence of linearized systems. This requires the introduction of a horizontal space constraint to ensure the local injectivity of the low-rank factorization. The method produces a sequence of approximate solutions for the original TV-SDP problem, for which we show that they stay close to the optimal solution path if properly initialized. Numerical experiments for a time-varying max-cut SDP relaxation demonstrate the computational advantages of the proposed method for tracking TV-SDPs in terms of runtime compared to off-the-shelf interior point methods.

翻译：我们提出了一种用于时变半定规划（TV-SDP）的在线算法，该算法基于路径跟踪过程中对低秩矩阵分解（即Burer-Monteiro分解）解轨迹的追踪。其中，预测-校正算法通过求解一系列线性化系统实现。为此需引入水平空间约束，以确保低秩分解的局部单射性。该方法为原始时变半定规划问题生成一系列近似解，我们证明：在初始化得当的条件下，这些近似解始终紧邻最优解路径。针对时变最大割半定规划松弛的数值实验表明，与现成的内点法相比，所提方法在跟踪时变半定规划时的计算效率具有显著优势。