A New Conjugate Gradient for Unconstrained Optimization Based on Step Size of Barzilai and Borwein
Keywords:
Unconstrained optimization, Conjugate gradient, Descent condition, Sufficient descent condition, Barzilai and Borwein step size, Global convergenceAbstract
In this paper, a new formula of is suggested for conjugate gradient method of solving unconstrained optimization problems based on step size of Barzilai and Borwein. Our new proposed CG-method has descent condition, sufficient descent condition and global convergence properties. Numerical comparisons with a standard conjugate gradient algorithm show that this algorithm very effective depending on the number of iterations and the number of functions evaluation.
References
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Barzilai, J. and Borwein, J.M. (1988), Tow point step size gradient methods, IMA J. Numer. Anal., 8, 141-148
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Fletcher, R., (1987), Practical methods of optimization, Unconstrained Optimization, John Wiley & Sons, New York, NY, USA.
Gilbert, J. C. and Nocedal, J. (1992), Global convergence properties of conjugate gradient methods for optimization, SIAM Journal Optimization, 2, 21–42.
Hestenes, M. R. and Stiefel, E. (1952), Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards. 49 , 409–436.
Liu, G. H.; Han, J. Y. and Yin, H. X. (1993), Global convergence of the Fletcher-Reeves algorithm with an inexact line search, Report, Institute of Applied Mathematics, Chinese Academy of Sciences.
Liu, Y. and Storey, C. (1991), Efficient generalized conjugate gradient algorithms, part 1: Theory, Journal of Optimization Theory and Applications, 69, 129-137.
Polak, E. and Ribiere, G. (1969), Note surla convergence des méthodes de directions conjuguées., 3(16), 35–43.
Polyak, B. T., (1969), The conjugate gradient method in extreme problems, USSR Comp. Math. and Math. Phys., 94-112.
Powell, M.J.D., (1977), Restart procedures for the conjugate gradient method, Mathematical Program. 12, 241–254.
Zoutendijk, G., (1970), Nonlinear Programming, Computational Methods in Integer and Nonlinear Programming. North-Holland Amsterdam, 37-86.
Barzilai, J. and Borwein, J.M. (1988), Tow point step size gradient methods, IMA J. Numer. Anal., 8, 141-148
Dai, Y. H. and Liao, L.Z. (2001), New conjugacy conditions and related nonlinear conjugate gradient methods, Application Mathematical Optimization, 43, 87–101.
Dai, Y. H. and Yuan, Y.(1995), Further studies on the Polak-Ribiere- Polyak method, Research report ICM-95-040, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences.
Dai, Y. H. and Yuan, Y. (1996), Convergence properties of the Fletcher-Reeves method, IMAJ. Numer. Anal., 2 , 155-164.
Dai, Y. H. and Yuan, Y. (1999), A nonlinear conjugate gradient method with a strong global convergence property, SIAM Journal on Optimization, 10 , 177-182.
Fletcher, R. and Reeves, C.M. (1964), Function minimization by conjugate gradients, The Computer Journal. 7 , 149–154.
Fletcher, R., (1987), Practical methods of optimization, Unconstrained Optimization, John Wiley & Sons, New York, NY, USA.
Gilbert, J. C. and Nocedal, J. (1992), Global convergence properties of conjugate gradient methods for optimization, SIAM Journal Optimization, 2, 21–42.
Hestenes, M. R. and Stiefel, E. (1952), Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards. 49 , 409–436.
Liu, G. H.; Han, J. Y. and Yin, H. X. (1993), Global convergence of the Fletcher-Reeves algorithm with an inexact line search, Report, Institute of Applied Mathematics, Chinese Academy of Sciences.
Liu, Y. and Storey, C. (1991), Efficient generalized conjugate gradient algorithms, part 1: Theory, Journal of Optimization Theory and Applications, 69, 129-137.
Polak, E. and Ribiere, G. (1969), Note surla convergence des méthodes de directions conjuguées., 3(16), 35–43.
Polyak, B. T., (1969), The conjugate gradient method in extreme problems, USSR Comp. Math. and Math. Phys., 94-112.
Powell, M.J.D., (1977), Restart procedures for the conjugate gradient method, Mathematical Program. 12, 241–254.
Zoutendijk, G., (1970), Nonlinear Programming, Computational Methods in Integer and Nonlinear Programming. North-Holland Amsterdam, 37-86.
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Published
2016-06-30
How to Cite
Shareef, S. G., & Ibrahim, A. L. (2016). A New Conjugate Gradient for Unconstrained Optimization Based on Step Size of Barzilai and Borwein. Science Journal of University of Zakho, 4(1), 104–114. Retrieved from https://sjuoz.uoz.edu.krd/index.php/sjuoz/article/view/311
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Science Journal of University of Zakho
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