TY - JOUR

T1 - Sparse semidefinite programs with guaranteed near-linear time complexity via dualized clique tree conversion

AU - Zhang, Richard Y.

AU - Lavaei, Javad

N1 - Funding Information:
This work was supported by the ONR YIP Award, DARPA YFA Award, AFOSR YIP Award, NSF CAREER Award, and ONR N000141712933.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2020

Y1 - 2020

N2 - Clique tree conversion solves large-scale semidefinite programs by splitting an n× n matrix variable into up to n smaller matrix variables, each representing a principal submatrix of up to ω× ω. Its fundamental weakness is the need to introduce overlap constraints that enforce agreement between different matrix variables, because these can result in dense coupling. In this paper, we show that by dualizing the clique tree conversion, the coupling due to the overlap constraints is guaranteed to be sparse over dense blocks, with a block sparsity pattern that coincides with the adjacency matrix of a tree. We consider two classes of semidefinite programs with favorable sparsity patterns that encompass the MAXCUT and MAX k-CUT relaxations, the Lovasz Theta problem, and the AC optimal power flow relaxation. Assuming that ω≪ n, we prove that the per-iteration cost of an interior-point method is linearO(n) time and memory, so an ϵ-accurate and ϵ-feasible iterate is obtained after O(nlog(1/ϵ)) iterations in near-linearO(n1.5log (1 / ϵ)) time. We confirm our theoretical insights with numerical results on semidefinite programs as large as n= 13 , 659.

AB - Clique tree conversion solves large-scale semidefinite programs by splitting an n× n matrix variable into up to n smaller matrix variables, each representing a principal submatrix of up to ω× ω. Its fundamental weakness is the need to introduce overlap constraints that enforce agreement between different matrix variables, because these can result in dense coupling. In this paper, we show that by dualizing the clique tree conversion, the coupling due to the overlap constraints is guaranteed to be sparse over dense blocks, with a block sparsity pattern that coincides with the adjacency matrix of a tree. We consider two classes of semidefinite programs with favorable sparsity patterns that encompass the MAXCUT and MAX k-CUT relaxations, the Lovasz Theta problem, and the AC optimal power flow relaxation. Assuming that ω≪ n, we prove that the per-iteration cost of an interior-point method is linearO(n) time and memory, so an ϵ-accurate and ϵ-feasible iterate is obtained after O(nlog(1/ϵ)) iterations in near-linearO(n1.5log (1 / ϵ)) time. We confirm our theoretical insights with numerical results on semidefinite programs as large as n= 13 , 659.

UR - http://www.scopus.com/inward/record.url?scp=85085357113&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85085357113&partnerID=8YFLogxK

U2 - 10.1007/s10107-020-01516-y

DO - 10.1007/s10107-020-01516-y

M3 - Article

AN - SCOPUS:85085357113

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

ER -