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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 320 (25 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.
Constraint Preconditioning for Indefinite Linear Systems
 SIAM J. Matrix Anal. Appl
, 2000
"... . The problem of nding good preconditioners for the numerical solution of indenite linear systems is considered. Special emphasis is put on preconditioners that have a 2 2 block structure and which incorporate the (1; 2) and (2; 1) blocks of the original matrix. Results concerning the spectrum and ..."
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Cited by 109 (14 self)
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. The problem of nding good preconditioners for the numerical solution of indenite linear systems is considered. Special emphasis is put on preconditioners that have a 2 2 block structure and which incorporate the (1; 2) and (2; 1) blocks of the original matrix. Results concerning the spectrum and form of the eigenvectors of the preconditioned matrix and its minimum polynomial are given. The consequences of these results are considered for a variety of Krylov subspace methods. Numerical experiments validate these conclusions. Key words. preconditioning, indenite matrices, Krylov subspace methods AMS subject classications. 65F10, 65F15, 65F50 1. Introduction. In this paper, we are concerned with investigating a new class of preconditioners for indenite systems of linear equations of a sort which arise in constrained optimization as well as in leastsquares, saddlepoint and Stokes problems. We attempt to solve the indenite linear system A B T B 0  {z } A x 1 x...
A preconditioner for generalized saddle point problems
 SIAM J. Matrix Anal. Appl
, 2004
"... Abstract. In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric/ skewsymmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matri ..."
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Cited by 41 (23 self)
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Abstract. In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric/ skewsymmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The potential of this approach is illustrated by numerical
Iterative solution of augmented systems arising in interior methods
 SIAM JOURNAL ON OPTIMIZATION
, 2007
"... Iterative methods are proposed for certain augmented systems of linear equations that arise in interior methods for general nonlinear optimization. Interior methods define a sequence of KKT equations that represent the symmetrized (but indefinite) equations associated with Newton’s method for a po ..."
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Cited by 20 (1 self)
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Iterative methods are proposed for certain augmented systems of linear equations that arise in interior methods for general nonlinear optimization. Interior methods define a sequence of KKT equations that represent the symmetrized (but indefinite) equations associated with Newton’s method for a point satisfying the perturbed optimality conditions. These equations involve both the primal and dual variables and become increasingly illconditioned as the optimization proceeds. In this context, an iterative linear solver must not only handle the illconditioning but also detect the occurrence of KKT matrices with the wrong matrix inertia. A oneparameter family of equivalent linear equations is formulated that includes the KKT system as a special case. The discussion focuses on a particular system from this family, known as the “doubly augmented system, ” that is positive definite with respect to both the primal and dual variables. This property means that a standard preconditioned conjugategradient method involving both primal and dual variables will either terminate successfully or detect if the KKT matrix has the wrong inertia. Constraint preconditioning is a wellknown technique for preconditioning the conjugategradient method on augmented systems. A family of constraint preconditioners is proposed that provably eliminates the inherent illconditioning in the augmented system. A considerable benefit of combining constraint preconditioning with the doubly augmented system is that the preconditioner need not be applied exactly. Two particular “activese ” constraint preconditioners are formulated that involve only a subset of the rows of the augmented system and thereby may be applied with considerably less work. Finally, some numerical experiments illustrate the numerical performance of the proposed preconditioners and highlight some theoretical properties of the preconditioned matrices.
Domain decomposition preconditioners for linear–quadratic elliptic optimal control problems
, 2004
"... ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquad ..."
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Cited by 19 (4 self)
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ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that for a class of problems the performance of the twolevel versions of our preconditioners is independent of the mesh size and of the subdomain size. 1.
Approximate factorization constraint preconditioners for saddlepoint matrices
 SIAM J. Sci. Comput
"... Abstract. We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Res ..."
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Cited by 19 (2 self)
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Abstract. We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions.
Nonmonotone Trust Region Methods for Nonlinear Equality Constrained Optimization without a Penalty Function
 MATH. PROGRAM., SER. B
, 2000
"... We propose and analyze a class of penaltyfunctionfree nonmonotone trustregion methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint viol ..."
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Cited by 16 (6 self)
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We propose and analyze a class of penaltyfunctionfree nonmonotone trustregion methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the ByrdOmojokun class of algorithms, each step is composed of a quasinormal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trustregion subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filterbased methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach.
Preconditioning KKT Systems
, 2002
"... This research presents new preconditioners for linear systems. We proceed from the most general case to the very specific problem area of sparse optimal control. In the first most general approach, we assume only that the coefficient matrix is nonsingular. We target highly indefinite, nonsymmetric p ..."
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Cited by 15 (0 self)
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This research presents new preconditioners for linear systems. We proceed from the most general case to the very specific problem area of sparse optimal control. In the first most general approach, we assume only that the coefficient matrix is nonsingular. We target highly indefinite, nonsymmetric problems that cause difficulties for preconditioned iterative solvers, and where standard preconditioners, like incomplete factorizations, often fail. We experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices. Our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing. Secondly, we present two new preconditioners for KKT systems. KKT systems arise in areas such as quadratic programming, sparse optimal control, and mixed finite element formulations. Our preconditioners approximate a constraint preconditioner with incomplete factorizations for the normal equations. Numerical experiments compare these two preconditioners with exact constraint preconditioning and the approach described above of permuting large entries to the diagonal. Finally, we turn to a specific problem area: sparse optimal control. Many optimal control problems are broken into several phases, and within a phase, most variables and constraints depend only on nearby variables and constraints. However, free initial and final times and timeindependent parameters impact variables and constraints throughout a phase, resulting in dense factored blocks in the KKT matrix. We drop fill due to these variables to reduce density within each phase. The resulting preconditioner is tightly banded and nearly block tridiagonal. Numerical experiments demonstrate that the preconditioners are effective, with very little fill in the factorization.
Structured preconditioners for nonsingular matrices of block twobytwo structures
 Math. Comp
"... Abstract. For the large sparse block twobytwo real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobitype, modified block Gaus ..."
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Cited by 15 (8 self)
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Abstract. For the large sparse block twobytwo real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobitype, modified block GaussSeideltype, and modified block unsymmetric (symmetric) GaussSeideltype preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block twobytwo coefficient matrices. In particular, these structured preconditioners can lead to efficient and highquality preconditioning matrices for some typical matrices from the realworld applications. 1.
Iterative methods for finding a trustregion step
, 2007
"... Abstract. We consider the problem of finding an approximate minimizer of a general quadratic function subject to a twonorm constraint. The SteihaugToint method minimizes the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constr ..."
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Cited by 14 (3 self)
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Abstract. We consider the problem of finding an approximate minimizer of a general quadratic function subject to a twonorm constraint. The SteihaugToint method minimizes the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constraint boundary. The benefit of this approach is that an approximate solution may be obtained with minimal work and storage. However, the method does not allow the accuracy of a constrained solution to be specified. We propose an extension of the SteihaugToint method that allows a solution to be calculated to any prescribed accuracy. If the SteihaugToint point lies on the boundary, the constrained problem is solved on a sequence of evolving lowdimensional subspaces. Each subspace includes an accelerator direction obtained from a regularized Newton method applied to the constrained problem. A crucial property of this direction is that it can be computed by applying the conjugategradient method to a positivedefinite system in both the primal and dual variables of the constrained problem. The method includes a parameter that allows the user to take advantage of the tradeoff between the overall number of function evaluations and matrixvector products associated with the underlying trustregion method. At one extreme, a lowaccuracy solution is obtained that is comparable to the SteihaugToint point. At the other extreme, a highaccuracy solution can be specified that minimizes the overall number of function evaluations at the expense of more matrixvector products. Key words. Largescale unconstrained optimization, trustregion methods, conjugategradient method, Lanczos tridiagonalization process AMS subject classifications. 49J20, 49J15, 49M37, 49D37, 65F05, 65K05, 90C30