Efficient Algorithm for Large-and-Sparse LMI Feasibility Problems

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) time and O(n) 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.