Efficient Algorithm for Large-and-Sparse LMI Feasibility Problems

Richard Y. Zhang, Javad Lavaei

Abstract: 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6.5</sup> ) time and O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) 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.