Decomposition Methods for Sparse Matrix Nearness Problems

We discuss three types of sparse matrix nearness problems: given a sparse symmetric matrix, find the matrix with the same sparsity pattern that is closest to it in Frobenius norm and (1) is positive semidefinite, (2) has a positive semidefinite completion, or (3) has a Euclidean distance matrix completion. Several proximal splitting and decomposition algorithms for these problems are presented and their performance is compared on a set of test problems. A key feature of the methods is that they involve a series of projections on small dense positive semidefinite or Euclidean distance matrix cones, corresponding to the cliques in a triangulation of the sparsity graph. The methods discussed include the dual block coordinate ascent algorithm (or Dykstra’s method), the dual projected gradient and accelerated projected gradient algorithms, and a primal and a dual application of the Douglas-Rachford splitting algorithm.