A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization

This letter presents a scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving unconstrained optimization problems. The basic idea is to combine the scaled memoryless BFGS method and the preconditioning technique in the frame of the conjugate gradient method. The preconditione...

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Vydáno v:	Applied mathematics letters Ročník 20; číslo 6; s. 645 - 650
Hlavní autor:	Andrei, Neculai
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Oxford Elsevier Ltd 01.06.2007 Elsevier
Témata:	BFGS preconditioning Conjugate gradient method Exact sciences and technology Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Sciences and techniques of general use Unconstrained optimization Conjugate gradient method Unconstrained optimization BFGS preconditioning Frame Gradient Spectral method Program Memory Optimization method Algorithm Direction Convergence Letter Applied mathematics Problem solving Metric Large scale Limit Preconditioning
ISSN:	0893-9659, 1873-5452
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Abstract	This letter presents a scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving unconstrained optimization problems. The basic idea is to combine the scaled memoryless BFGS method and the preconditioning technique in the frame of the conjugate gradient method. The preconditioner, which is also a scaled memoryless BFGS matrix, is reset when the Powell restart criterion holds. The parameter scaling the gradient is selected as the spectral gradient. Computational results for a set consisting of 750 test unconstrained optimization problems show that this new scaled conjugate gradient algorithm substantially outperforms known conjugate gradient methods such as the spectral conjugate gradient SCG of Birgin and Martínez [E. Birgin, J.M. Martínez, A spectral conjugate gradient method for unconstrained optimization, Appl. Math. Optim. 43 (2001) 117–128] and the (classical) conjugate gradient of Polak and Ribière [E. Polak, G. Ribière, Note sur la convergence de méthodes de directions conjuguées, Revue Francaise Informat. Reserche Opérationnelle, 3e Année 16 (1969) 35–43], but subject to the CPU time metric it is outperformed by L-BFGS [D. Liu, J. Nocedal, On the limited memory BFGS method for large scale optimization, Math. Program. B 45 (1989) 503–528; J. Nocedal. http://www.ece.northwestern.edu/~nocedal/lbfgs.html].
AbstractList	This letter presents a scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving unconstrained optimization problems. The basic idea is to combine the scaled memoryless BFGS method and the preconditioning technique in the frame of the conjugate gradient method. The preconditioner, which is also a scaled memoryless BFGS matrix, is reset when the Powell restart criterion holds. The parameter scaling the gradient is selected as the spectral gradient. Computational results for a set consisting of 750 test unconstrained optimization problems show that this new scaled conjugate gradient algorithm substantially outperforms known conjugate gradient methods such as the spectral conjugate gradient SCG of Birgin and Martínez [E. Birgin, J.M. Martínez, A spectral conjugate gradient method for unconstrained optimization, Appl. Math. Optim. 43 (2001) 117–128] and the (classical) conjugate gradient of Polak and Ribière [E. Polak, G. Ribière, Note sur la convergence de méthodes de directions conjuguées, Revue Francaise Informat. Reserche Opérationnelle, 3e Année 16 (1969) 35–43], but subject to the CPU time metric it is outperformed by L-BFGS [D. Liu, J. Nocedal, On the limited memory BFGS method for large scale optimization, Math. Program. B 45 (1989) 503–528; J. Nocedal. http://www.ece.northwestern.edu/~nocedal/lbfgs.html].
Author	Andrei, Neculai
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Keywords	Conjugate gradient method Unconstrained optimization BFGS preconditioning Frame Gradient Spectral method Program Memory Optimization method Algorithm Direction Convergence Letter Applied mathematics Problem solving Metric Large scale Limit Preconditioning
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SubjectTerms	BFGS preconditioning Conjugate gradient method Exact sciences and technology Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Sciences and techniques of general use Unconstrained optimization
Title	A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization
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