Exploring chordal sparsity in semidefinite programming with sparse plus low-rank data matrices

TL;DR

This paper investigates SDP problems with sparse plus low-rank data matrices, developing a framework to convert them into easier-to-solve sparse SDPs with bounded tree-width and establishing tight rank bounds.

Contribution

It introduces a unified framework for converting SPLR-structured SDPs into sparse SDPs with bounded tree-width and derives tight rank bounds for these problems.

Findings

Conversion framework for SPLR SDPs to sparse SDPs

Derivation of tight rank bounds in worst-case scenarios

Enhanced understanding of structure in large-scale SDPs

Abstract

Semidefinite programming (SDP) problems are challenging to solve because of their high dimensionality. However, solving sparse SDP problems with small tree-width are known to be relatively easier because: (1) they can be decomposed into smaller multi-block SDP problems through chordal conversion; (2) they have low-rank optimal solutions. In this paper, we study more general SDP problems whose coefficient matrices have sparse plus low-rank (SPLR) structure. We develop a unified framework to convert such problems into sparse SDP problems with bounded tree-width. Based on this, we derive rank bounds for SDP problems with SPLR structure, which are tight in the worst case.