🤖 AI Summary
This work addresses the optimization landscape analysis of the Burer–Monteiro (BM) low-rank factorization method for MaxCut-type semidefinite programs (SDPs). We identify the condition number of the associated Laplacian matrix as a key spectral criterion: when this condition number falls below a tight, explicitly characterized threshold, every second-order critical point of the nonconvex BM formulation is guaranteed to be a global optimum, and no spurious local minima exist. This constitutes the first tight sufficient condition ensuring global convergence of the BM method for MaxCut-type SDPs, markedly improving the theoretical solvability boundary for canonical problems such as ℤ₂-synchronization. Technically, we integrate tools from Riemannian optimization, spectral graph theory, and second-order critical point analysis to establish a precise quantitative relationship between the Laplacian condition number and the benignness of the optimization landscape. Our results provide a more rigorous theoretical foundation for solving nonconvex low-rank SDPs.
📝 Abstract
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.