Efficient Algorithm for Large-and-Sparse LMI Feasibility Problems

Richard Y. Zhang, Javad Lavaei

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Linear matrix inequalities (LMIs) play a fundamental role in robust and optimal control theory. However, their practical use remains limited, in part because their solution complexities of O(n-{6.5}) time and O(n-{4}) memory limit their applicability to systems containing no more than a few hundred state variables. This paper describes a Newton-PCG algorithm to efficiently solve large-and-sparse LMI feasibility problems, based on efficient log-det barriers for sparse matrices. Assuming that the data matrices share a sparsity pattern that admits a sparse Cholesky factorization, we prove that the algorithm converges in linear O(n) time and memory. The algorithm is highly efficient in practice: we solve LMI feasibility problems over power system models with as many as n=5738 state variables in 2 minutes on a standard workstation running MATLAB.

Original languageEnglish (US)
Title of host publication2018 IEEE Conference on Decision and Control, CDC 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages8
ISBN (Electronic)9781538613955
StatePublished - Jan 18 2019
Externally publishedYes
Event57th IEEE Conference on Decision and Control, CDC 2018 - Miami, United States
Duration: Dec 17 2018Dec 19 2018

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0743-1546


Conference57th IEEE Conference on Decision and Control, CDC 2018
Country/TerritoryUnited States

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization


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