Chordal decomposition in operator-splitting methods for sparse semidefinite programs

TL;DR

This paper introduces a chordal decomposition approach for large sparse SDPs, enabling scalable first-order methods like ADMM to efficiently solve decomposed problems and identify infeasibility.

Contribution

It develops a novel chordal decomposition technique for SDPs that facilitates the application of operator-splitting methods, including an extended ADMM algorithm for primal-dual problems.

Findings

Significant computational improvements over existing solvers.

Ability to identify infeasible problems and generate certificates.

Efficient parallelizable projections onto small PSD cones.

Abstract

We employ chordal decomposition to reformulate a large and sparse semidefinite program (SDP), either in primal or dual standard form, into an equivalent SDP with smaller positive semidefinite (PSD) constraints. In contrast to previous approaches, the decomposed SDP is suitable for the application of first-order operator-splitting methods, enabling the development of efficient and scalable algorithms. In particular, we apply the alternating direction method of multipliers (ADMM) to solve decomposed primal- and dual-standard-form SDPs. Each iteration of such ADMM algorithms requires a projection onto an affine subspace, and a set of projections onto small PSD cones that can be computed in parallel. We also formulate the homogeneous self-dual embedding (HSDE) of a primal-dual pair of decomposed SDPs, and extend a recent ADMM-based algorithm to exploit the structure of our HSDE. The resulting HSDE algorithm has the same leading-order computational cost as those for the primal or dual problems only, with the advantage of being able to identify infeasible problems and produce an infeasibility certificate. All algorithms are implemented in the open-source MATLAB solver CDCS. Numerical experiments on a range of large-scale SDPs demonstrate the computational advantages of the proposed methods compared to common state-of-the-art solvers.