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Iterative Methods for IllConditioned Linear Systems From Optimization
, 1998
"... Preconditioned conjugategradient methods are proposed for solving the illconditioned linear systems which arise in penalty and barrier methods for nonlinear minimization. The preconditioners are chosen so as to isolate the dominant cause of ill conditioning. The methods are stablized using a restr ..."
Abstract

Cited by 6 (2 self)
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Preconditioned conjugategradient methods are proposed for solving the illconditioned linear systems which arise in penalty and barrier methods for nonlinear minimization. The preconditioners are chosen so as to isolate the dominant cause of ill conditioning. The methods are stablized using a restricted form of iterative refinement. Numerical results illustrate the approaches considered. 1 Email : n.gould@rl.ac.uk 2 Current reports available from "http://www.rl.ac.uk/departments/ccd/numerical/reports/reports.html". Department for Computation and Information Atlas Centre Rutherford Appleton Laboratory Oxfordshire OX11 0QX August 26, 1998. 1 INTRODUCTION 1 1 Introduction Let A and H be, respectively, fullrank m by n (m n) and symmetric n by n real matrices. Suppose furthermore that any nonzero coefficients in this data are modest, that is the data is O(1). (1) We consider the iterative solution of the linear system (H +A T D \Gamma1 A)x = b (1.1) where b is modest an...
A modified Newton method for minimization
 Journal of Optimization Theory and Applications
, 1977
"... Abstract. Some promising ideas for minimizing a nonlinear function, whose first and second derivatives are given, by a modified Newton method, were introduced by Fiacco and McCormick (Ref. 1). Unfortunately, in developing a method around these ideas, Fiacco and McCormick used a potentially unstable ..."
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Cited by 4 (0 self)
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Abstract. Some promising ideas for minimizing a nonlinear function, whose first and second derivatives are given, by a modified Newton method, were introduced by Fiacco and McCormick (Ref. 1). Unfortunately, in developing a method around these ideas, Fiacco and McCormick used a potentially unstable, or even impossible, matrix factorization. Using some recently developed techniques for factorizing an indefinite symmetric matrix, we are able to produce a method which is similar to Fiacco and ~cCormick's original method, but avoids the difficulties of the original method. Key Words. Modified Newton methods, negative curvature directions, unconstrained minimization. 1.
Constructing Appropriate Models for LargeScale, LinearlyConstrained, Nonconvex, Nonlinear Optimization Algorithms
, 1995
"... We consider the algebraic issues concerning the solution of general, largescale, linearly constrained nonlinear optimization problems. Particular attention is given to suitable methods for solving the linear systems which occur at each iteration of such methods. The main issue addressed is how to e ..."
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Cited by 3 (1 self)
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We consider the algebraic issues concerning the solution of general, largescale, linearly constrained nonlinear optimization problems. Particular attention is given to suitable methods for solving the linear systems which occur at each iteration of such methods. The main issue addressed is how to ensure that a quadratic model of the objective function is positive definite in the nullspace of the constraints, while not adversely affecting the convergence of Newton's method nor incurring a significant computational overhead. Numerical evidence to support the theoretical developments is provided.
Towards a Parallel Tile LDL Factorization for Multicore Architectures
, 2011
"... The increasing number of cores in modern architectures requires the development of new algorithms as a means to achieving concurrency and hence scalability. This paper presents an algorithm to compute the LDL T factorization of symmetric indefinite matrices without taking pivoting into consideration ..."
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Cited by 1 (1 self)
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The increasing number of cores in modern architectures requires the development of new algorithms as a means to achieving concurrency and hence scalability. This paper presents an algorithm to compute the LDL T factorization of symmetric indefinite matrices without taking pivoting into consideration. The algorithm, based on the factorizations presented by Buttari et al. [11], represents operations as a sequence of small tasks that operate on square blocks of data. These tasks can be scheduled for execution based on dependencies among them and on computational resources available. This allows an out of order execution of tasks that removes the intrinsically sequential nature of the factorization. Numerical and performance results are presented. Numerical results were limited to matrices for which pivoting is not numerically necessary. A performance comparison between LDL T, Cholesky and LU factorizations and the performance of the kernels required by LDL T, which are an extension of level3 BLAS kernels, are presented.