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A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization. (English) Zbl 1116.90114
Summary: 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 E. Birgin and J. M. Martínez [Appl. Math. Optim. 43, No. 2, 117–128 (2001; Zbl 0990.90134)] and the (classical) conjugate gradient of E. Polak and G. Ribière [Rev. Franç. Inform. Rech. Opér. 3, No. 16, 35–43 (1969; Zbl 0174.48001)], but subject to the CPU time metric it is outperformed by L-BFGS [D. C. Liu and J. Nocedal, Math. Program., Ser. B 45, No. 3, 503–528 (1989; Zbl 0696.90048)]; J. Nocedal, http://www.ece.northwestern.edu/ nocedal/lbfgs.html].
MSC:
90C52Methods of reduced gradient type
90C30Nonlinear programming
65K05Mathematical programming (numerical methods)
Software:
L-BFGS