We consider MaxCut-type semidefinite programs (SDP) which admit a low rank solution. To numerically leverage the low rank hypothesis, a standard algorithmic approach is the Burer-Monteiro factorization, which allows to significantly reduce the dimensionality of the problem at the cost of its convexity. We give a sharp condition on the conditioning of the Laplacian matrix associated with the SDP under which any second-order critical point of the non-convex problem is a global minimizer. By applying our theorem, we improve on recent results about the correctness of the Burer-Monteiro approach on $\mathbb{Z}_2$-synchronization problems.

翻译：我们研究一类具有低秩解的MaxCut型半定规划（SDP）。为在数值计算中利用低秩假设，标准的算法途径是采用Burer-Monteiro分解，该方法能以牺牲凸性为代价显著降低问题的维度。我们给出了与该SDP相关的拉普拉斯矩阵条件数的一个精确条件，在此条件下非凸问题的任何二阶临界点都是全局极小点。通过应用我们的定理，我们改进了近期关于Burer-Monteiro方法在$\mathbb{Z}_2$-同步问题上正确性的相关结果。