jzs-10822

A New Spectral on the Gradient Methods for Unconstrained Optimization Minimization Problem

Basim A.  Hassan1 and Hawraz N. Jabbar*2

1Department of Mathematics, College of Computers Sciences and Mathematics, University of Mosul, Iraq.

2Department of Mathematics, College of Sciences, University of Kirkuk, Iraq.

*Corresponding author’s e-mail: hawrazmath@gmail.com

Original: 3 February 2020        Revised: 21 August 2020        Accepted: 26 September 2020        Published online: 20 December 2020

Doi Link:

Abstract

One simple and well-known method for minimizing the functions is a spectral conjugate gradient method. In this paper, we derive a new spectral on the  method of gradient, which can give a new path of search. The new spectral method holds the property of descent and we have shown that the spectral method is convergent globally. The empirical results show that for the test problems, the given approach is competitive with the other conjugate gradient methods.

Key Words: Unconstrained Optimization, Conjugate Gradient Methods, Large Scale Methods

References

[1] Andrei N. "An Unconstrained Optimization test function collection". Adv. Model. Optimization. Vol. 10, pp.147-161. (2008).

[2] Basim A. Hassan, "A new formula for conjugate parameter computation based on the quadratic model". Indonesian Journal of Electrical Engineering and Computer Science. Vol. 3, pp. 954-961. (2019).

[3] Basim A. Hassan. "A Globally Convergence Spectral Conjugate Gradient Method for Solving Unconstrained Optimization Problems and Mohammed". Raf. J. of Comp. & Math’s. Vol. 10, pp. 21-28. (2013).

[4] Basim A. Hassan. "Development a Special Conjugate Gradient Algorithm for Solving Unconstrained Minimization Problems". Raf. J. of Comp. & Math’s. Vol. 9, pp. 73-84. (2012).

[5] Dai, Y.  H. and Yuan, Y. "A nonlinear conjugate gradient method with a strong global convergence property". SIAM  J. optimization. pp. 177-182. (1999).

[6] Fletcher, R. "Practical Method of Optimization". 2nd Edition. John Wiley and Sons. New York. (1989).

[7] Fletcher, R. and Reeves, C. M. "Funtion minimization by conjagate gradients". Computer Journal. Vol. 7, pp. 149-154. (1964).

[8] Hager and W.W. and Zhang H. "A new conjugate gradient method with guaranteed descent and an efficient line search". SIAM J. Optim. Vol. 16, pp. 170–192. (2005).

[9] Hu C. M.  and Wan Z. "An extended spectral conjugate gradient method for unconstrained optimization problems". British Journal of Mathematics & Computer Science. Vol. 3, No. 2, pp.86-98. (2013).

[10] Hestenes, M. R. and Stiefel, E. "Methods of conjugate gradients for solving linear systems". Journal of Research of National Burean of Standard. Vol. 49, pp. 409-436. (1952).

[11] Liu, Y. and Storey, C. "Efficient generalized conjugate gradients algorithms". Part 1: Theory. J. Optimization Theory and Applications. Vol. 69, pp. 129-137. (1991).

[12] Luksan, L., Matonoha, C. and Vlcek J. "Computational experience with conjugate gradient  methods for unconstrained optimization". Technical Report. No.1038, pp. 1-17. (2008).

[13] Ming L., Hongwei L. and Zexian L. "A new family of conjugate gradient methods for unconstrained optimization". J. Appl. Math. Comput. (2017).

[14] Narushima Y. and Yae H. "A survey of sufficient descent conjugate gradient methods for unconstrained optimization". SUT J. Math. Vol. 50, pp. 167–203. (2014).

[15] Polak, E. and Ribiere, G. "Note sur la Convergence de Directions Conjugate". Revue Francaise Informant. Reserche. Opertionelle. Vol. 16, pp. 35-43. (1969).

[16] Rao S. "Engineering Optimization Theory and Practice". 4th edition, John Wiley & Sons Inc., New Jersey, Canada. (2009).

[17] Zhongbo S., Hongyang L., Jing W. and  Yantao T. "Two modified spectral conjugate gradient methods and their global convergence for unconstrained optimization". International Journal of Computer Mathematics, (2017).

[18] Zoutendijk, G. "Nonlinear programming, computational methods". (In: Abadie, J. (eds.) Integerand Nonlinear Programming' . North-Holland, Amsterdam) .pp. 37–86. (1970).

View Full Article