Fast ADMM for Semidefinite Programs with Chordal Sparsity

TL;DR

This paper introduces an efficient first-order ADMM-based method for solving large-scale semidefinite programs with chordal sparsity, significantly improving computational speed by exploiting problem structure.

Contribution

It develops scalable ADMM algorithms utilizing chordal decomposition for sparse SDPs, applicable to both primal and dual forms, with efficient projections and open-source implementation.

Findings

Achieves speedups over existing SDP solvers on large sparse problems.

Demonstrates effectiveness on SDPLIB and random block-arrow sparse SDPs.

Provides scalable solutions for control theory applications.

Abstract

Many problems in control theory can be formulated as semidefinite programs (SDPs). For large-scale SDPs, it is important to exploit the inherent sparsity to improve the scalability. This paper develops efficient first-order methods to solve SDPs with chordal sparsity based on the alternating direction method of multipliers (ADMM). We show that chordal decomposition can be applied to either the primal or the dual standard form of a sparse SDP, resulting in scaled versions of ADMM algorithms with the same computational cost. Each iteration of our algorithms consists of a projection on the product of small positive semidefinite cones, followed by a projection on an affine set, both of which can be carried out efficiently. Our techniques are implemented in CDCS, an open source add-on to MATLAB. Numerical experiments on large-scale sparse problems in SDPLIB and random SDPs with block-arrow sparse patterns show speedups compared to some common state-of-the-art software packages.