Benign landscape for Burer-Monteiro factorizations of MaxCut-type semidefinite programs

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 and the Kuramoto model.

@article{endor2025_2411.03103,
  title={ Benign landscape for Burer-Monteiro factorizations of MaxCut-type semidefinite programs },
  author={ Faniriana Rakoto Endor and Irène Waldspurger },
  journal={arXiv preprint arXiv:2411.03103},
  year={ 2025 }
}