On the convergence of conjugate gradient algorithms

Pytlak, R.

doi:10.1093/imanum/14.3.443

Cited by 18 publications

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“…In the paper we consider a version of conjugate gradient algorithm which was introduced in [26]. Its direction finding subproblem is given by…”

Section: Introductionmentioning

confidence: 99%

“…Notice, that if β k ≡ 1 then we have the Lemaréchal-Wolfe algorithm proposed in [20] and [31]. In [26] it was shown that the Lemaréchal-Wolfe algorithm is in fact the Fletcher-Reeves algorithm when directional minimization is exact. Moreover, the sequence {β k } was constructed in such a way that directions generated by (3) are equivalent to those provided by the Polak-Ribière formula (under the assumption that directional minimization is exact).…”

Section: Introductionmentioning

confidence: 99%

“…The convergence analysis discussed in Sect. 2 borrows from the relevant parts of [27] and [28] (see also [26]). We state, under some assumptions on scaling matrices, the globally convergent Polak-Ribière version of our general algorithm.…”

Section: Introductionmentioning

confidence: 99%

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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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“…In the paper we consider a version of conjugate gradient algorithm which was introduced in [26]. Its direction finding subproblem is given by…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Preconditioned conjugate gradient algorithms for nonconvex problems with box constraints

Pytlak

Tarnawski

2010

Numer. Math.

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“…In [9] it was shown that the Wolfe-LemarCchal algorithm is in fact the Fletcher-Reeves algorithm when directional minimization is exact. Moreover, the sequence { P k ) was constructed in such way that directions generated by (3) are equivalent to those provided by the Polak-RibiCre formula (under the assumption that directional minimization is exact).…”

Section: /I Is the Euclidean Norm And Gr = V F ( X K )mentioning

confidence: 99%

“…In [9] (see also [4]) a new family of conjugate gradient algorithms has been introduced whose direction finding subproblem is given by where N r { a , b ) is defined as the point from a line segment spanned by the vectors a and b which has the smallest norm, i.e.,…”

Section: Introductionmentioning

confidence: 99%

Preconditioned Conjugate Gradient Algorithms for Nonconvex Problems with Box Constraints

Pytlak

Tarnawski

IFIP International Federation for Information Processing

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The paper describes a new conjugate gradient algorithm for large scale nonconvex problems with box constraints. In order to speed up the convergence the algorithm employs a scaling matrix which transforms the space of original variables into the space in which Hessian matrices of functionals describing the problems have more clustered eigenvalues. This is done efficiently by applying limited memory BFGS updating matrices. Once the scaling matrix is calculated, the next few iterations of the conjugate gradient algorithms are performed in the transformed space. The box constraints are treated by the projection as previously used in [R.Pytlak, The efficient algorithm for large-scale problems with simple bounds on the variables, SIAM J. on Optimization, Vol. 8, 532-560, 19981. We believe that the preconditioned conjugate gradient algorithm gives more flexibility in achieving balance between the computing time and the number of function evaluations in comparison to a limited memory BFGS algorithm. The numerical results show that the proposed method is competitive to L-BFGS-B procedure.keywords: bound constrained nonlinear optimization problems, conjugate gradient algorithms, quasi-Newton methods.

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Nonlinear Conjugate Gradient Methods

Dai¹

2011

Wiley Encyclopedia of Operations Research and Management Science

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Conjugate gradient methods are a class of important methods for solving linear equations and for solving nonlinear optimization. In this article, a review on conjugate gradient methods for unconstrained optimization is given. They are divided into early conjugate gradient methods, descent conjugate gradient methods, and sufficient descent conjugate gradient methods. Two general convergence theorems are provided for the conjugate gradient method assuming the descent property of each search direction. Some research issues on conjugate gradient methods are mentioned.

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On the convergence of conjugate gradient algorithms

Cited by 18 publications

References 0 publications

Preconditioned conjugate gradient algorithms for nonconvex problems with box constraints

Preconditioned conjugate gradient algorithms for nonconvex problems with box constraints

Preconditioned Conjugate Gradient Algorithms for Nonconvex Problems with Box Constraints

Nonlinear Conjugate Gradient Methods

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