Burer-Monteiro ADMM for Large-scale SDPs

TL;DR

This paper introduces a bilinear decomposition combined with ADMM for large-scale SDPs, reducing computational complexity and achieving global convergence with empirical success in reaching the optimum.

Contribution

It presents a novel bilinear decomposition for the Burer-Monteiro method and combines it with ADMM, enabling efficient solutions for large-scale SDPs with convergence guarantees.

Findings

Outperforms state-of-the-art Riemannian algorithms

Converges globally to a first-order stationary point

Empirically reaches the global optimum

Abstract

We propose a bilinear decomposition for the Burer-Monteiro method and combine it with the standard Alternating Direction Method of Multipliers algorithm for semidefinite programming. Bilinear decomposition reduces the degree of the augmented Lagrangian from four to two, which makes each of the subproblems a quadratic programming and hence computationally efficient. Our approach is able to solve a class of large-scale SDPs with diagonal constraints. We prove that our ADMM algorithm converges globally to a first-order stationary point, and show by exploiting the negative curvature that the algorithm converges to a point within $O(1-1/r)$ of the optimal objective value. Additionally, the proximal variant of the algorithm can solve block-diagonally constrained SDPs with global convergence to a first-order stationary point. Numerical results show that both our ADMM algorithm and the proximal variant outperform the state-of-art Riemannian manifold algorithms and can reach the global optimum empirically.