Fast ADMM for homogeneous self-dual embedding of sparse SDPs

TL;DR

This paper introduces a fast ADMM-based method for solving sparse semidefinite programs using homogeneous self-dual embedding, achieving efficiency and solution certificates, with implementation in an open-source solver.

Contribution

The paper presents a novel ADMM algorithm that exploits chordal sparsity and block elimination to efficiently solve homogeneous self-dual embeddings of SDPs, providing primal, dual, and infeasibility certificates.

Findings

Achieves similar per-iteration cost as splitting methods on primal or dual.

More efficient than existing first-order methods for sparse conic programs.

Demonstrates speed-ups on benchmark problems from SDPLIB.

Abstract

We propose an efficient first-order method, based on the alternating direction method of multipliers (ADMM), to solve the homogeneous self-dual embedding problem for a primal-dual pair of semidefinite programs (SDPs) with chordal sparsity. Using a series of block eliminations, the per-iteration cost of our method is the same as applying a splitting method to the primal or dual alone. Moreover, our approach is more efficient than other first-order methods for generic sparse conic programs since we work with smaller semidefinite cones. In contrast to previous first-order methods that exploit chordal sparsity, our algorithm returns both primal and dual solutions when available, and a certificate of infeasibility otherwise. Our techniques are implemented in the open-source MATLAB solver CDCS. Numerical experiments on three sets of benchmark problems from the library SDPLIB show speed-ups compared to some common state-of-the-art software packages.