Preconditioned conjugate gradient algorithms for nonconvex problems

Pytlak, R.; Tarnawski, Tomasz

doi:10.1109/cdc.2004.4608810

Cited by 7 publications

(20 citation statements)

References 28 publications

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“…If the surface is convex, no local minimal are present and minimization techniques generally provide the point position with good accuracy. In contrast, in the analysis of concave surfaces, convergence problems can appear due to the presence of local minimal [15,16]. This is an especially important problem in the analysis of systems with multiple interactions (combinations of reflections and diffractions) that consist of both convex and concave surfaces.…”

Section: Introductionmentioning

confidence: 99%

Application of the Particle Swarm Optimization to the Computation of the Go/Utd Reflection Points Over Nurbs Surfaces

Gutiérrez¹,

Adana²,

Valiente³

2012

PIER Letters

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Abstract-A new method to find the Geometrical Optics/Uniform Theory of Diffraction reflection points over Non Uniform Rational BSplines surfaces is presented. The approach is based on the Particle Swarm Optimization (PSO) technique, and the cost function used to find the reflection points is based on Snell's law. The technique can be used as an alternative to classic minimization techniques in cases where convergence problems arise.

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Section: Introductionmentioning

confidence: 99%

Application of the Particle Swarm Optimization to the Computation of the Go/Utd Reflection Points Over Nurbs Surfaces

Gutiérrez¹,

Adana²,

Valiente³

2012

PIER Letters

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“…[29] ). The latter schemes also include algorithms where the Nonlinear Conjugate Gradient method is often integrated with an ad hoc preconditioner, and is coupled with a linesearch procedure to compute the steplength α k .…”

Section: Global Convergence For An Effective Pncgmentioning

confidence: 99%

“…Here we first recall a general scheme of a Preconditioned Nonlinear Conjugate Gradient (PNCG) algorithm (see also [29] for a general survey). Then, we detail the description of the rationale behind our proposal, namely the idea of using low-rank updates to satisfy a secant-like equation.…”

Section: General Remarks On Pncgmentioning

confidence: 99%

“…Observe that addressing good preconditioners for NCG methods still remains an intriguing research issue, in particular when the solution of large scale problems is sought and no structure of the problem is known in advance [6,9,12,21,29,30] . Similarly, matrix-free preconditioning for linear systems or sequences of linear systems is currently an appealing research topic, too (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Among them, the NCG along with quasi-Newton methods are undoubtedly the most commonly used, since they both proved to be really effective in practice and show mature convergence properties (see e.g. [28,29] ).…”

Section: Introductionmentioning

confidence: 99%

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An adaptive truncation criterion, for linesearch-based truncated Newton methods in large scale nonconvex optimization

Caliciotti

Fasano

Nash

et al. 2018

Operations Research Letters

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a b s t r a c tThis paper includes a twofold result for the Nonlinear Conjugate Gradient (NCG) method, in large scale unconstrained optimization. First we consider a theoretical analysis, where preconditioning is embedded in a strong convergence framework of an NCG method from the literature. Mild conditions to be satisfied by the preconditioners are defined, in order to preserve NCG convergence. As a second task, we also detail the use of novel matrix-free preconditioners for NCG. Our proposals are based on quasi-Newton updates, and either satisfy the secant equation or a secant-like condition at some of the previous iterates. We show that, in some sense, the preconditioners we propose also approximate the inverse of the Hessian matrix. In particular, the structures of our preconditioners depend on lowrank updates used, along with different choices of specific parameters. The low-rank updates are obtained as by-product of NCG iterations. The results of an extended numerical experience using large scale CUTEst problems is reported, showing that our preconditioners can considerably improve the performance of NCG methods.

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Preconditioned conjugate gradient algorithms for nonconvex problems with box constraints

Pytlak

Tarnawski

2010

Numer. Math.

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The paper describes new conjugate gradient algorithms for large scale nonconvex problems with box constraints. In order to speed up convergence the algorithms employ scaling matrices which transform the space of original variables into the space in which Hessian matrices of the problem's functionals have more clustered eigenvalues. This is done by applying limited memory BFGS updating matrices. Once the scaling matrix is calculated, the next few conjugate gradient iterations are performed in the transformed space. The box constraints are treated efficiently by the projection. We also present a limited memory quasi-Newton method which is a special version of our general algorithm. The presented algorithms have strong global convergence properties, in particular they identify constraints active at a solution in a finite number of iterations. We believe that they are competitive to the L-BFGS-B method and present some numerical results which support our claim.

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Preconditioned conjugate gradient algorithms for nonconvex problems

Cited by 7 publications

References 28 publications

Application of the Particle Swarm Optimization to the Computation of the Go/Utd Reflection Points Over Nurbs Surfaces

Application of the Particle Swarm Optimization to the Computation of the Go/Utd Reflection Points Over Nurbs Surfaces

An adaptive truncation criterion, for linesearch-based truncated Newton methods in large scale nonconvex optimization

Preconditioned conjugate gradient algorithms for nonconvex problems with box constraints

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