A NEW HYBRID PRP-MMSIS CONJUGATE GRADIENT METHOD AND ITS APPLICATION IN PORTOFOLIO SELECTION

Main Article Content

Sindy Devila
Maulana Malik
Wed Giyarti

Abstract

In this paper, we propose a new hybrid coefficient of conjugate gradient method (CG) for solving unconstrained optimization model.  The new coefficient is combination of part the MMSIS (Malik et.al, 2020) and PRP (Polak, Ribi'ere \& Polyak, 1969) coefficients.  Under exact line search, the search direction of new method satisfies the sufficient descent condition and based on certain assumption, we establish the global convergence properties.  Using some test functions, numerical results show that the proposed method is more efficient than MMSIS method.  Besides, the new method can be used to solve problem in minimizing portfolio selection risk .

Article Details

Section
Algebra
Author Biographies

Sindy Devila, Universitas Indonesia

Department of Mathematics

Maulana Malik, Universitas Indonesia

Department of Mathematics

Wed Giyarti, Universitas Islam Negeri Sunan Kalijaga Yogyakarta

Program Studi Pendidikan Matematika

References

[1] J. Nocedal and S. Wright,Numerical optimization. Springer Science & Business Media, 2006.

[2] M. R. Hestenes and E. Stiefel, œMethods of conjugate gradients for solving linear systems,Journal of research of the National Bureau of Standards, vol. 49, no. 6, pp. 409436, 1952.

[3] E. Polak and G. Ribiere, œNote sur la convergence de m ́ethodes de directions conjugu ́ees,Revuefranc ̧aise dinformatique et de recherche op ́erationnelle. S ́erie rouge, vol. 3, no. 16, pp. 3543,1969.

[4] B. T. Polyak, œThe conjugate gradient method in extremal problems,USSR ComputationalMathematics and Mathematical Physics, vol. 9, no. 4, pp. 94112, 1969.

[5] Y. Liu and C. Storey, œEfficient generalized conjugate gradient algorithms, part 1: theory,Jour-nal of optimization theory and applications, vol. 69, no. 1, pp. 129137, 1991.

[6] R. Fletcher and C. M. Reeves, œFunction minimization by conjugate gradients,The computerjournal, vol. 7, no. 2, pp. 149154, 1964.

[7] R. Fletcher,Practical methods of optimization. John Wiley & Sons, 2013.

[8] Y. Dai and Y. X. Yuan, œA nonlinear conjugate gradient method with a strong global convergenceproperty,SIAM Journal on optimization, vol. 10, no. 1, pp. 177182, 1999.

[9] D. Touati-Ahmed and C. Storey, œEfficient hybrid conjugate gradient techniques,Journal ofoptimization theory and applications, vol. 64, no. 2, pp. 379397, 1990.

[10] Y. Hu and C. Storey, œGlobal convergence result for conjugate gradient methods,Journal ofOptimization Theory and Applications, vol. 71, no. 2, pp. 399405, 1991.

[11] J. C. Gilbert and J. Nocedal, œGlobal convergence properties of conjugate gradient methods foroptimization,SIAM Journal on optimization, vol. 2, no. 1, pp. 2142, 1992.

[12] Y.-h. Dai and Y. Yuan, œAn efficient hybrid conjugate gradient method for unconstrained opti-mization,Annals of Operations Research, vol. 103, no. 1, pp. 3347, 2001.

[13] M. Malik, M. Mamat, S. S. Abas, I. M. Sulaiman, and Sukono, œA new coefficient of the conju-gate gradient method with the sufficient descent condition and global convergence properties.,Engineering Letters, vol. 28, no. 3, pp. 704714, 2020.

[14] L. Zhang, œAn improved weiyaoliu nonlinear conjugate gradient method for optimizationcomputation,Applied Mathematics and computation, vol. 215, no. 6, pp. 22692274, 2009.

 

[15] M. Malik, M. Mamat, S. S. Abas, I. M. Sulaiman, A. T. Bon,et al., œComparison of conju-gate gradient method on solving unconstrained optimization problems, inProceedings of theInternational Conference on Industrial Engineering and Operations Management, no. August,2020.

[16] M. Malik, M. Mamat, S. S. Abas,et al., œConvergence analysis of a new coefficient conju-gate gradient method under exact line search,International Journal of Advanced Science andTechnology, vol. 29, no. 5, pp. 187198, 2020.

[17] M. Malik, M. Mamat, S. S. Abas, I. M. Sulaiman,et al., œA new modification of nprp conjugategradient method for unconstrained optimization,Advances in Mathematics: Scientific Journal,vol. 9, no. 7, pp. 49554970, 2020.

[18] M. Malik, S. S. Abas, M. Mamat, I. S. Mohammed,et al., œA new hybrid conjugate gradientmethod with global convergence properties,International Journal of Advanced Science andTechnology, vol. 29, no. 5, pp. 199210, 2020.

[19] M. Malik, M. Mamat, S. S. Abas, I. M. Sulaiman,et al., œA new spectral conjugate gradientmethod with descent condition and global convergence property for unconstrained optimiza-tion,J. Math. Comput. Sci., vol. 10, no. 5, pp. 20532069, 2020.

[20] M. Malik, M. Mamat, S. S. Abas, I. M. Sulaiman, A. T. Bon,et al., œSolving unconstrainedminimization problems with a new hybrid conjugate gradient method, inProceedings of theInternational Conference on Industrial Engineering and Operations Management, no. August,2020.

[21] G. Zoutendijk, œNonlinear programming, computational methods,Integer and nonlinear pro-gramming, pp. 3786, 1970.

[22] N. Andrei,Nonlinear Conjugate Gradient Methods for Unconstrained Optimization. Springer,2020.

[23] M. Jamil and X.-S. Yang, œA literature survey of benchmark functions for global optimisa-tion problems,International Journal of Mathematical Modelling and Numerical Optimisation,vol. 4, no. 2, pp. 150194, 2013.

[24] E. D. Dolan and J. J. Mor ́e, œBenchmarking optimization software with performance profiles,Mathematical Programming, vol. 91, no. 2, pp. 201213, 2002.

[25] S. Roman,Introduction to the mathematics of finance: from risk management to options pricing.Springer Science & Business Media, 2004.