Linear solvers for power grid optimization problems: a review of GPU-accelerated linear solvers

The linear equations that arise in interior methods for constrained optimization are sparse symmetric indefinite and become extremely ill-conditioned as the interior method converges. These linear systems present a challenge for existing solver frameworks based on sparse LU or LDL^T decompositions. We benchmark five well known direct linear solver packages using matrices extracted from power grid optimization problems. The achieved solution accuracy varies greatly among the packages. None of the tested packages delivers significant GPU acceleration for our test cases

A Schur complement approach to preconditioning sparse linear least-squares problems with some dense rows

The effectiveness of sparse matrix techniques for directly solving large-scale linear least-squares problems is severely limited if the system matrix A has one or more nearly dense rows. In this paper, we partition the rows of A into sparse rows and dense rows (A s and A d ) and apply the Schur complement approach. A potential difficulty is that the reduced normal matrix AsTA s is often rank-deficient, even if A is of full rank. To overcome this, we propose explicitly removing null columns of A s and then employing a regularization parameter and using the resulting Cholesky factors as a preconditioner for an iterative solver applied to the symmetric indefinite reduced augmented system. We consider complete factorizations as well as incomplete Cholesky factorizations of the shifted reduced normal matrix. Numerical experiments are performed on a range of large least-squares problems arising from practical applications. These demonstrate the effectiveness of the proposed approach when combined with either a sparse parallel direct solver or a robust incomplete Cholesky factorization algorithm

On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Mor´e and Sorensen, methods which obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including in the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems

On using Cholesky-based factorizations and regularization for solving rank-deficient sparse linear least-squares problems

By examining the performance of modern parallel sparse direct solvers and exploiting our knowledge of the algorithms behind them, we perform numerical experiments to study how they can be used to efficiently solve rank-deficient sparse linear least-squares problems arising from practical applications. The Cholesky factorization of the normal equations breaks down when the least-squares problem is rank-deficient, while applying a symmetric indefinite solver to the augmented system can give an unacceptable level of fill in the factors. To try to resolve these difficulties, we consider a regularization procedure that modifies the diagonal of the unregularized matrix. This leads to matrices that are easier to factorize. We consider both the regularized normal equations and the regularized augmented system. We employ the computed factors of the regularized systems as preconditioners with an iterative solver to obtain the solution of the original (unregularized) problem. Furthermore, we look at using limited-memory incomplete Cholesky-based factorizations and how these can offer the potential to solve very large problems

Decomposition in conic optimization with partially separable structure

Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables. However in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower-dimensional cones. The most important example is sparse semidefinite programming with a chordal sparsity pattern. Here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive-semidefinite-completable matrices with chordal sparsity patterns. The paper describes a decomposition method that exploits partial separability in conic linear optimization. The method is based on Spingarn's method for equality constrained convex optimization, combined with a fast interior-point method for evaluating proximal operators

The state-of-the-art of preconditioners for sparse linear least-squares problems

In recent years, a variety of preconditioners have been proposed for use in solving large sparse linear least-squares problems. These include simple diagonal preconditioning, preconditioners based on incomplete factorizations and stationary inner iterations used with Krylov subspace methods. In this study, we briefly review preconditioners for which software has been made available and then present a numerical evaluation of them using performance profiles and a large set of problems arising from practical applications. Comparisons are made with state-of-the-art sparse direct methods

Combinatorial problems in solving linear systems

42 pages, available as LIP research report RR-2009-15Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices

High-resolution imaging for e-heritage

Ph.DDOCTOR OF PHILOSOPH