A modified spectral conjugate gradient method for solving unconstrained minimization problems

A. A. Danhausa, R. M. Odekunle, A. S. Onanaye

Abstract


The development a modified spectral conjugate gradient method for solving unconstrained minimization problem is considered in this paper. A new Conjugate (update) parameter is obtained by the idea of Dai-Kou's technique for generating conjugate parameters. A new spectral parameter is also presented based on quasi-Newton direction and quasi-Newton condition. Under the strong Wolfe line search, the proposed method is proved to be globally convergent. Numerical results showed that the algorithm takes lesser number of iteration to obtain the minimum of a given function compared to the method Loannis and Panagiotis’s spectral CGM [36]  (LP for short), Birgin and Martinez’s  spectral CGM [2] (BM for short), and Jinbao, Qian, Xianzhen, Youfang and Jianghua’s spectral CGM [31]  ( JQXYJ for short). We hereby recommend this method for the solution of both linear and Non-linear Unconstrained optimization problem.

Keywords: Unconstrained minimization problem, Spectral conjugate gradient method, Global convergence, numerical results.


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References


(1) N. Andrei, New Accelerated Conjugate Gradient Algorithms as a Mo Modification of Dai-Yuan's Computational scheme for Unconstrained Opimization, J. Comput. Appl. Math., 234, 3397-3410, 2010.

(2) E. G Birgin and J. M. Martinez, A Spectral Conjugate Gradient Method for Unconstrained Optimization, Applied Mathematics and Optimization, 43,117-128, 1999

(3) J. Barzilai and J. M. Borwien, Two Point Step Gradient Methods, IMA J. Numer. Anal., 8, 141-148, 1988.

(4) Y. H. Dai and C. X. Kou, A Nonlinear Conjugate Gradient Algorithm with an Optimal Property and an improved Wolfe line Search, SIAM J. Optim., 23, 296-320, 2013.

(5) Y. H. Dai and Y. Yuan, A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property, SIAM Journal on Optimization, 10, 177-182, 1999.

(6) E. D. Dolan and J. J. Mor'e, Benchmarking Optimization Software with Perfomance profiles, Math. Program. 91, 201-213, 2002.

(7) R. Fletcher, Low Storage Methods for Unconstrained Optimization, Lectures in Applied Mathematics, American Mathematical Society, Providence, RI, 26, 165-179, 1990.

(8) R. Fletcher and C. M. Reeves, Functional Minimization by Conjugate Gradients, Comput. J., 7, 149-154, 1964.

(9) M. R. Hestenes and E. Stiefel, Method of Conjugate Gradient for solving Linear System, J. Res. Nat. Bur. Standards, 49, 409-436, 1952.

(10) J. Jinbao, C. Qian, J. Xianzhen, Z. Youfang and Y. Jianghua, A new Spectral Conjugate Gradient Method for Large-scale Unconstrained Optimization, Optimization Methods and Software, DOI:10:1080/10556788.2016.1225213, 2016.

(11) E. L. Loannis and P Panagiotis, A new Spectral Conjugate Gradient Methods based om a Modified Secant Equation for Unconstrained Optimization, Elsevier, 396-405, 2012.

(12) Y. Liu and C. Storey, E¢ cient Generalized Conjugate Gradient Algorithms, part : Theory, J. Optim. Theory Apppl. 69, 129-137, 1991.

(13) E. Polak and G. Ribie're, Note Sur la convergence de directions conjuge'es, Rev. Fr. Inform. Rech. Oper. 16, 35-43, 1969.


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