A clique graph based merging strategy for decomposable SDPs

TL;DR

This paper introduces a novel clique merging strategy based on clique graphs to improve the efficiency of solving decomposable SDPs, demonstrating performance gains over existing methods.

Contribution

A new clique merging approach utilizing clique graphs is proposed, enhancing the reduction of SDP solve times compared to prior techniques.

Findings

The method reduces solve time in benchmark problems.

It outperforms existing clique merging techniques.

Implemented in the COSMO solver for practical use.

Abstract

Chordal decomposition techniques are used to reduce large structured positive semidefinite matrix constraints in semidefinite programs (SDPs). The resulting equivalent problem contains multiple smaller constraints on the nonzero blocks (or cliques) of the original problem matrices. This usually leads to a significant reduction in the overall solve time. A further reduction is possible by remerging cliques with significant overlap. The degree of overlap for which this is effective is dependent on the particular solution algorithm and hardware to be employed. We propose a novel clique merging approach that utilizes the clique graph to identify suitable merge candidates. We show its performance by comparing it with two existing methods on selected problems from a benchmark library. Our approach is implemented in the latest version of the conic ADMM-solver COSMO.