Complexity of chordal conversion for sparse semidefinite programs with small treewidth

If a sparse semidefinite program (SDP), specified over n×n matrices and subject to m linear constraints, has an aggregate sparsity graph G with small treewidth, then chordal conversion will sometimes allow an interior-point method to solve the SDP in just O(m+n) time per-iteration, which is a signif...

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Bibliographic Details
Published in:Mathematical programming Vol. 213; no. 1-2; pp. 201 - 237
Main Author: Zhang, Richard Y.
Format: Journal Article
Language:English
Published: Heidelberg Springer Nature B.V 01.09.2025
Subjects:
Constraints
Localization
Optimization
Polynomials
Power flow
Sensors
Sparsity
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:If a sparse semidefinite program (SDP), specified over n×n matrices and subject to m linear constraints, has an aggregate sparsity graph G with small treewidth, then chordal conversion will sometimes allow an interior-point method to solve the SDP in just O(m+n) time per-iteration, which is a significant speedup over the Ω(n3) time per-iteration for a direct application of the interior-point method. Unfortunately, the speedup is not guaranteed by an O(1) treewidth in G that is independent of m and n, as a diagonal SDP would have treewidth zero but can still necessitate up to Ω(n3) time per-iteration. Instead, we construct an extended aggregate sparsity graph G¯⊇G by forcing each constraint matrix Ai to be its own clique in G. We prove that a small treewidth in G¯ does indeed guarantee that chordal conversion will solve the SDP in O(m+n) time per-iteration, to ϵ-accuracy in at most O(m+nlog(1/ϵ)) iterations. This sufficient condition covers many successful applications of chordal conversion, including the MAX-k-CUT relaxation, the Lovász theta problem, sensor network localization, polynomial optimization, and the AC optimal power flow relaxation, thus allowing theory to match practical experience.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-024-02137-5