Chordal Graphs and Semidefinite Optimization

Abstract

Chordal graphs play a central role in techniques for exploiting sparsityin large semidefinite optimization problems and in related convexoptimization problems involving sparse positive semidefinite matrices.Chordal graph properties are also fundamental to several classicalresults in combinatorial optimization, linear algebra, statistics,signal processing, machine learning, and nonlinear optimization. Thissurvey covers the theory and applications of chordal graphs, with anemphasis on algorithms developed in the literature on sparse Choleskyfactorization. These algorithms are formulated as recursions on eliminationtrees, supernodal elimination trees, or clique trees associatedwith the graph. The best known example is the multifrontal Choleskyfactorization algorithm, but similar algorithms can be formulated fora variety of related problems, including the computation of the partialinverse of a sparse positive definite matrix, positive semidefinite andEuclidean distance matrix completion problems, and the evaluation ofgradients and Hessians of logarithmic barriers for cones of sparse positivesemidefinite matrices and their dual cones. The purpose of thesurvey is to show how these techniques can be applied in algorithmsfor sparse semidefinite optimization, and to point out the connectionswith related topics outside semidefinite optimization, such as probabilisticnetworks, matrix completion problems, and partial separabilityin nonlinear optimization.