The Augmented Mixing Method: Computing High-Accuracy Primal-Dual Solutions to Large-Scale SDPs via Column Updates

Abstract: The Burer-Monteiro factorization has become a powerful tool for solving large-scale semidefinite programs (SDPs), enabling recently developed low-rank solvers to tackle problems previously beyond reach. However, existing methods are typically designed to prioritize scalability over solution accuracy. We introduce the Augmented Mixing Method, a new algorithm that combines the Burer-Monteiro factorization with an inexact augmented Lagrangian framework and a block coordinate descent scheme. Our method emphasizes solving the resulting subproblems efficiently and to high precision. Inequality constraints are handled directly, without reformulation or introducing slack variables. A novel dynamic update strategy for the penalty parameter ensures that primal and dual feasibility progress remain balanced. This approach enables our method to compute highly accurate primal-dual solutions, even for large-scale SDPs with more than ten million inequality constraints. Despite lacking theoretical convergence guarantees, the Augmented Mixing Method shows strong practical performance with default parameters, across a wide range of SDP instances. It often produces more accurate primal-dual solutions than state-of-the-art interior-point methods and scales significantly better. Our open-source Julia implementation is memory-efficient, customizable, and supports arbitrary-precision arithmetic.