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Efficient hybrid conjugate gradient techniques. (English) Zbl 0666.90063
Descent properties and global convergence proofs are given for a new hybrid conjugate gradient algorithm. Computational results for this algorithm are also given and compared with those of the Fletcher-Reeves and the Polak-Ribière methods, showing a considerable improvement over the latter two methods. We also give new criteria for restarting conjugate gradient algorithms that prove to be computationally very efficient. These criteria provide a descent property and global convergence for any conjugate gradient algorithm using a nonnegative update $\beta$.
Reviewer: D.Touati-Ahmed
##### MSC:
 90C30 Nonlinear programming 65K05 Mathematical programming (numerical methods) 90C52 Methods of reduced gradient type 49M37 Methods of nonlinear programming type in calculus of variations
##### References:
 [1] Fletcher, R.,Practical Methods of Optimization, Vol. 1, Unconstrained Optimization, Wiley, Chichester, England, 1980. [2] Goldstein, A. A.,On Steepest Descent. SIAM Journal on Control, Vol. 3, pp. 147-151, 1965. [3] Fletcher, R., andReeves, C. M.,Function Minimization by Conjugate Gradients, Computer Journal, Vol. 7, pp. 143-154, 1964. · Zbl 0132.11701 · doi:10.1093/comjnl/7.2.149 [4] Polak, E., andRibière, G.,Note sur la Convergence des Méthodes de Directions Conjuguées, Revue Française, Information et Recherche Opérationelle, Vol. 16, pp. 35-43, 1969. [5] Powell, M. J. D.,Nonconvex Minimization Calculations and the Conjugate Gradient Method, Report No. DAMTP 1983/NA14, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England, 1983. [6] Al-Baali, M.,Descent Property and Global Convergence of the Fletcher-Reeves Method with Inexact Line-Search, IMA Journal of Numerical Analysis, Vol. 5, pp. 121-124, 1985. · Zbl 0578.65063 · doi:10.1093/imanum/5.1.121 [7] Powell, M. J. D.,Restart Procedures for the Conjugate Gradient Method, Mathematical Programming, Vol. 12, pp. 241-254, 1977. · Zbl 0396.90072 · doi:10.1007/BF01593790 [8] Powell, M. J. D.,Convergence Properties of Algorithms for Nonlinear Optimization, Report No. DAMTP 1985/NA1, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England, 1985. [9] Shanno, D. F.,Globally Convergent Conjugate Gradient Algorithms, Mathematical Programming, Vol. 33, pp. 61-67, 1985. · Zbl 0579.90079 · doi:10.1007/BF01582011 [10] Touati-Ahmed, D., andStorey, C.,Globally Convergent Hybrid Conjugate Gradient Methods, Mathematics Research Report No. 196, Department of Mathematics, Loughborough University of Technology, Loughborough, Leicestershire, England, 1986. [11] Wolfe, M. A.,Numerical Methods for Unconstrained Optimization: An Introduction, Van Nostrand Reinhold, London, England, 1978. [12] Touati-Ahmed, D.,Efficient Hybrid Conjugate-Gradient Techniques, PhD Thesis, Department of Mathematics, Loughborough University of Technology, Loughborough, Leicestershire, England, 1988.