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

TL;DR

The paper introduces the Augmented Mixing Method, a scalable algorithm combining Burer-Monteiro factorization with an augmented Lagrangian approach, achieving high-accuracy primal-dual solutions for large-scale SDPs with millions of constraints.

Contribution

It presents a novel algorithm that efficiently solves large-scale SDPs with high accuracy, directly handles inequality constraints, and balances primal-dual feasibility without theoretical convergence guarantees.

Findings

Achieves high-accuracy solutions for SDPs with over ten million constraints.

Outperforms state-of-the-art interior-point methods in accuracy and scalability.

Open-source Julia implementation supports arbitrary-precision arithmetic.

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.