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75
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 180 (30 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
Implementation of Interior Point Methods for Large Scale Linear Programming
 in Interior Point Methods in Mathematical Programming
, 1996
"... In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on bot ..."
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Cited by 70 (22 self)
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In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on both sides. The significant difference between interior point and simplex based methods is reflected not only in the theoretical background but also in the practical implementation. In this paper we give an overview of the most important characteristics of advanced implementations of interior point methods. First, we present the infeasibleprimaldual algorithm which is widely considered the most efficient general purpose IPM. Our discussion includes various algorithmic enhancements of the basic algorithm. The only shortcoming of the "traditional" infeasibleprimaldual algorithm is to detect a possible primal or dual infeasibility of the linear program. We discuss how this problem can be solve...
Primaldual interior methods for nonconvex nonlinear programming
 SIAM Journal on Optimization
, 1998
"... Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterize ..."
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Cited by 59 (5 self)
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Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penaltybarrier function that involves both primal and dual variables. Each subproblem is solved with a modified Newton method that generates search directions from a primaldual system similar to that proposed for interior methods. The augmented penaltybarrier function may be interpreted as a merit function for values of the primal and dual variables. An inertiacontrolling symmetric indefinite factorization is used to provide descent directions and directions of negative curvature for the augmented penaltybarrier merit function. A method suitable for large problems can be obtained by providing a version of this factorization that will treat large sparse indefinite systems.
A column preordering strategy for the unsymmetricpattern multifrontal method
 ACM Transactions on Mathematical Software
, 2004
"... A new method for sparse LU factorization is presented that combines a column preordering strategy with a rightlooking unsymmetricpattern multifrontal numerical factorization. The column ordering is selected to give a good a priori upper bound on fillin and then refined during numerical factoriza ..."
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Cited by 56 (4 self)
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A new method for sparse LU factorization is presented that combines a column preordering strategy with a rightlooking unsymmetricpattern multifrontal numerical factorization. The column ordering is selected to give a good a priori upper bound on fillin and then refined during numerical factorization (while preserving the bound). Pivot rows are selected to maintain numerical stability and to preserve sparsity. The method analyzes the matrix and automatically selects one of three preordering and pivoting strategies. The number of nonzeros in the LU factors computed by the method is typically less than or equal to those found by a wide range of unsymmetric sparse LU factorization methods, including leftlooking methods and prior multifrontal methods.
Symmetric quasidefinite matrices
 SIAM Journal on Optimization
, 1995
"... We say that a symmetric matrix K is quasidefinite if it has the form ..."
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Cited by 54 (3 self)
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We say that a symmetric matrix K is quasidefinite if it has the form
BILUM: Block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems
 SIAM J. Sci. Comput
, 1999
"... Abstract. We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typ ..."
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Cited by 53 (29 self)
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Abstract. We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typically enjoyed by multigrid methods. Several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed. The advantages of block ILUM over point ILUM include increased robustness and efficiency. We compare several versions of the block ILUM, point ILUM, and the dualthresholdbased ILUT preconditioners. In particular, tests with some convectiondiffusion problems show that it may be possible to obtain convergence that is nearly independent of the Reynolds number as well as of the grid size.
A QMRbased interiorpoint algorithm for solving linear programs
 Math. Programming
, 1994
"... A new approach for the implementation of interiorpoint methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2\Theta2block systems of linear equations that arise within the interiorpoint algorithm. These linear systems ..."
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Cited by 39 (4 self)
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A new approach for the implementation of interiorpoint methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2\Theta2block systems of linear equations that arise within the interiorpoint algorithm. These linear systems are solved by a symmetric variant of the quasiminimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3 \Theta 3block systems to symmetric 2 \Theta 2block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interiorpoint algorithm is proposed. Some indefini...
A New KrylovSubspace Method For Symmetric Indefinite Linear Systems
, 1994
"... Many important applications involve the solution of large linear systems with symmetric, but indefinite coefficient matrices. For example, such systems arise in incompressible flow computations and as subproblems in optimization algorithms for linear and nonlinear programs. Existing Krylovsubspace ..."
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Cited by 35 (0 self)
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Many important applications involve the solution of large linear systems with symmetric, but indefinite coefficient matrices. For example, such systems arise in incompressible flow computations and as subproblems in optimization algorithms for linear and nonlinear programs. Existing Krylovsubspace iterations for symmetric indefinite systems, such as SYMMLQ and MINRES, require the use of symmetric positive definite preconditioners, which is a rather unnatural restriction when the matrix itself is highly indefinite with both many positive and many negative eigenvalues. In this note, we describe a new Krylovsubspace iteration for solving symmetric indefinite linear systems that can be combined with arbitrary symmetric preconditioners. The algorithm can be interpreted as a special case of the quasiminimal residual method for general nonHermitian linear systems, and like the latter, it produces iterates defined by a quasiminimal residual property. The proposed method has the same work ...
Software for simplified Lanczos and QMR algorithms
 Appl. Numer. Math
, 1995
"... The nonsymmetric Lanczos process simplifies when applied to Jsymmetric and JHermitian matrices, and work and storage requirements are roughly halved compared to the general case. In this paper, we describe FORTRAN77 implementations of simplified versions of the lookahead Lanczos algorithm and o ..."
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Cited by 35 (6 self)
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The nonsymmetric Lanczos process simplifies when applied to Jsymmetric and JHermitian matrices, and work and storage requirements are roughly halved compared to the general case. In this paper, we describe FORTRAN77 implementations of simplified versions of the lookahead Lanczos algorithm and of the quasiminimal residual (QMR) method, which is a Lanczosbased iterative procedure for the solution of linear systems. These implementations of the simplified algorithms complete our software package QMRPACK, which so far contained only codes for Lanczos and QMR algorithms for general matrices. We describe in some detail the use of two routines, one for the solution of linear systems, and the other for eigenvalue computations. We present examples that lead to Jsymmetric and JHermitian matrices. Results of numerical experiments are reported. Keywords. Lanczos process; quasiminimal residual iteration; linear system; eigenvalue computation; Jsymmetric matrix; JHermitian matrix; look...
An interior algorithm for nonlinear optimization that combines line search and trust region steps
 Mathematical Programming 107
, 2006
"... An interiorpoint method for nonlinear programming is presented. It enjoys the flexibility of switching between a line search method that computes steps by factoring the primaldual equations and a trust region method that uses a conjugate gradient iteration. Steps computed by direct factorization a ..."
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Cited by 31 (11 self)
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An interiorpoint method for nonlinear programming is presented. It enjoys the flexibility of switching between a line search method that computes steps by factoring the primaldual equations and a trust region method that uses a conjugate gradient iteration. Steps computed by direct factorization are always tried first, but if they are deemed ineffective, a trust region iteration that guarantees progress toward stationarity is invoked. To demonstrate its effectiveness, the algorithm is implemented in the Knitro [6, 28] software package and is extensively tested on a wide selection of test problems. 1