Nonsmooth Equation Method for Nonlinear Nonconvex Optimization

Lukšan, Ladislav; Matonoha, Ctirad; Vlček, Jan

doi:10.1007/978-3-642-18560-1_8

Cited by 5 publications

(17 citation statements)

References 5 publications

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“…Such preconditioners are widely used in linear systems obtained from a discretization of partial differential equations [15,24]. The preconditioners for the augmented system have also been used in the context of linear programming [14,23] and in the context of nonlinear programming [11,19,21,22,25]. As was shown in [23], the preconditioners for indefinite augmented system offer more freedom than those for the normal equations.…”

Section: Preconditionersmentioning

confidence: 99%

“…Matrix P of (3) belongs to a wide class of block-preconditioners that have been studied in numerous applications in the context of partial differential equations, see for example [15,24] and the references therein, and in the context of optimization [11,14,19,21,22]. We perform the spectral analysis of P −1 H and conclude, as in [19], that for convex optimization problems all the eigenvalues of this matrix are strictly positive.…”

Section: Introductionmentioning

confidence: 99%

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Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Bergamaschi

Gondzio

Zilli

2004

Computational Optimization and Applications

120

128

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Abstract. Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.

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Section: Preconditionersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Bergamaschi

Gondzio

Zilli

2004

Computational Optimization and Applications

120

128

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“…If it is used, the preconditioned conjugate gradient method can be efficiently applied to (8), even if matrix K is indefinite. This fact follows from the three theorems below, which are proved in [19] (we restrict to the situation when the matrixĜ −D is nonsingular, which is a common situation and also the worst case in some sense, see [16]). Notice that the preconditioned conjugate gradient method can be written in the following algorithmic form.…”

Section: −1mentioning

confidence: 90%

“…This contribution contains a survey of results proved in [19], new results concerning the use of filters described in [4] and a new algorithm based on these results. This algorithm was tested and compared with the algorithm proposed in [19] by using three collections of large scale nonlinear programming problems.…”

Section: Primal Formulationmentioning

confidence: 99%

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Interior point methods for large-scale nonlinear programming

Lukšan

Matonoha

Vlček

2005

Optimization Methods and Software

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Abstract:In the contribution, we describe an algorithm for solving nonlinear nonconvex programming problems, which is based on the interior point approach. The main theoretical results concern direction determination and step-length selection. We split inequality constraints into active and inactive parts to overcome problems with instability. Inactive constraints are eliminated directly, while active constraints are used for defining a symmetric indefinite linear system. Inexact solution of this system is obtained iteratively using indefinitely preconditioned conjugate gradient method. Theorems confirming efficiency of the indefinite preconditioner are introduced. Furthermore, a new merit function is defined and a filter principle is used for step-length selection. The algorithm was implemented in the interactive system for universal functional optimization UFO. Results of numerical experiments are reported.

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On the solution of indefinite systems arising in nonlinear programming problems

Bonettini

Ruggiero

Tinti

2007

Numerical Linear Algebra App

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This work is concerned with the convergence properties and the numerical analysis of the preconditioned conjugate gradient (PCG) method applied to the solution of indefinite linear systems arising in nonlinear optimization. Our approach is based on the choice of quasidefinite preconditioners and of a suitable factorization routine. Some theoretical and numerical results about these preconditioners are obtained. Furthermore, we show the behaviour of the PCG method for different formulations of the indefinite system and we compare the effectiveness of the proposed variants

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Nonsmooth Equation Method for Nonlinear Nonconvex Optimization

Cited by 5 publications

References 5 publications

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Interior point methods for large-scale nonlinear programming

On the solution of indefinite systems arising in nonlinear programming problems

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