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

Tianyun Tang, Kim-Chuan Toh

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.

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

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.

Paper Structure

This paper contains 21 sections, 11 theorems, 83 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Linear SDP problem
  3. Sparse SDP and chordal conversion
  4. SDP with sparse plus low-rank structure
  5. Contribution 1: sparse extension
  6. Contribution 2: rank bound
  7. Outline of the paper
  8. Preliminaries on tree-width, path-width and binary tree
  9. Sparse extension of SDP with SPLR structure
  10. A simple example
  11. Construction of sparse extension for (\ref{['SDP']})
  12. Proof of Theorem \ref{['theo:spex']}
  13. Chordal conversion of sparse extension
  14. The rank bound of SDP with SPLR structure
  15. Proof of Theorem \ref{['ubthmtree']}.
  16. ...and 6 more sections

Key Result

Theorem 1.4

Suppose the problem (SDP) has $(G,\ell)-$SPLR structure for some $\ell\in \mathbb{N}$. Let $$T,V$$ be a tree-decomposition of $G$ such that $|T|=p$ and its width wid($T,{\cal V}$) = tw(G). Then it has a sparse extension (SDPex), whose precise form is given in newprob, with sparsity pattern $\widehat

Figures (11)

  • Figure 1: Chordal graph $G=([12],\mathcal{E})$ with ${\rm tw}(G)=3$. The maximal cliques are $\{1,6,7\},\{1,2,8\},\{2,3,9\},\{3,4,10\},\{4,5,11\},\{5,6,12\},\{1,2,5,6\},\{2,3,4,5\}.$
  • Figure 2: Clique tree of the chordal graph in Figure \ref{['fig:chordal graph']}.
  • Figure 3: Splitting vertex $x$ with degree 4 into a path with four vertices $x_1,x_2,x_3,x_4.$ Each of its neighbours is connected to one vertex on the path.
  • Figure 4: A splitting of the clique tree in Figure \ref{['fig:clique tree']} with respect to $\{1,2,5,6\}.$ What we obtain is a new clique tree decomposition with the same width and 1 fewer vertex of degree greater than $3.$
  • Figure 5: A splitting of the clique tree decomposition in Figure \ref{['fig:split1']} with respect to $\{2,3,4,5\}.$ What we obtain is a binary clique tree decomposition with width 3, which is the tree-width of the graph in Figure \ref{['fig:chordal graph']}.
  • ...and 6 more figures

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 28 more