Degeneracy in Maximal Clique Decomposition for Semidefinite Programs

TL;DR

This paper investigates the degeneracy issues in maximal clique decomposition of SDPs, revealing conditions leading to non-uniqueness of multipliers and increased condition numbers, which impact solution stability.

Contribution

It identifies primal degeneracy and non-uniqueness conditions in clique-based SDP decompositions, highlighting their effects on numerical stability and solution quality.

Findings

Maximal clique decomposition is primal degenerate for low-rank SDPs.

Conditions for non-uniqueness of multipliers are derived.

Decomposition leads to higher condition numbers in the Schur complement matrix.

Abstract

Exploiting sparsity in Semidefinite Programs (SDP) is critical to solving large-scale problems. The chordal completion based maximal clique decomposition is the preferred approach for exploiting sparsity in SDPs. In this paper, we show that the maximal clique-based SDP decomposition is primal degenerate when the SDP has a low rank solution. We also derive conditions under which the multipliers in the maximal clique-based SDP formulation is not unique. Numerical experiments demonstrate that the SDP decomposition results in the schur-complement matrix of the Interior Point Method (IPM) having higher condition number than for the original SDP formulation.