International Journal of Nonlinear Analysis and Applications

An improved PRP conjugate gradient method for optimization computation

Document Type : Research Paper

Authors

1 Badji Mokhtar University, Annaba, 23000, Algeria

2 Laboratory Informatics and Mathematics (LiM), Mohamed Cherif Messaadia University, Souk Ahras, 41000, Algeria

3 Superior School of Industrial Technologies, Annaba, 23000, Algeria

Abstract

The conjugate gradient method plays a very important role in several fields, to solve problems of large sizes. To improve the efficiency of this method, a lot of work has been done; in this paper, we propose a new modification of PRP method to solve a large scale unconstrained optimization problems in relation with strong Wolf Powell Line Search property, when the latter was used under some conditions, a global convergence result was proved. In comparison with other known methods the efficiency of this method proved that it is better in the number of iterations and in time on $90$ proposed problems by use of Matlab.

Keywords

[1] M. Al-Baali, Descent property and global convergence of the Fletcher—Reeves method with inexact line search, IMA J. Numer. Anal. 5 (1985), no. 1, 121–124.
[2] N. Andrei, An unconstrained optimization test functions collection, Adv. Model. Optim 10 (2008), no. 1, 147–161.
[3] E. Blum, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), 123–145.
[4] F.H. Clarke, Y.S. Ledyaev, R.J. Stern, and P.R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178, Springer Science & Business Media, 2008.
[5] R. Fletcher and C.M. Reeves, Function minimization by conjugate gradients, Comput. J 7 (1964), no. 2, 149–154.
[6] J.C. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM J. Optim. 2 (1992), no. 1, 21–42.
[7] M. Hamoda, M. Mamat, M. Rivaie, and Z. Salleh, A conjugate gradient method with strong Wolfe-Powell line search for unconstrained optimization, Appl. Math. Sci. 10 (2016), 721–734.
[8] M.R Hestenes and E. Stiefel, Methods of conjugate gradients for solving, J. Res. Nat. Bureau Stand. 49 (1952), no. 6, 409.
[9] D.E. Knuth, The TEXbook, Addison Wesley Professional, Massachusetts, 1984.
[10] G. Li, C. Tang, and Z.Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math. 202 (2007), no. 2, 523–539.
[11] Y. Liu and C. Storey, Efficient generalized conjugate gradient algorithms, part 1: theory, J. Optim. Theory Appl. 69 (1991), no. 1, 129–137.
[12] Mu. Mamat, M. Rivaie, I. Mohd, and M. Fauzi, A new conjugate gradient coefficient for unconstrained optimization, Int. J. Contemp. Math. Sci. 5 (2010), no. 29, 1429–1437.
[13] I.S. Mohammed, M. Mamat, A. Abashar, M. Rivaie, and Z. Salleh, A modified nonlinear conjugate gradient method for unconstrained optimization, Appl. Math. Sci. 9 (2015), no. 54, 2671–2682.
[14] T. Nguyen Xuan and T. Phan Nhat, On the existence of equilibrium points of vector functions, Numer. Funct. Anal. Optim. 19 (1998), no. 1-2, 141–156.
[15] O. Omer, M. Mamat, and M. Rivaie, The global convergence properties of a family of conjugate gradient method under the strong Wolfe line search, Abstr. Appl. Anal., vol. 2015, 2015.
[16] M.J.D. Powell, Restart procedures for the conjugate gradient method, Math. Program. 12 (1977), no. 1, 241–254.
[17] G. Quon, S. Haider, A.G. Deshwar, A. Cui, P.C. Boutros, and Q. Morris, Computational purification of individual tumor gene expression profiles leads to significant improvements in prognostic prediction, Genome Med. 5 (2013), no. 3, 1–20.
[18] M. Rivaie, A. Abashar, M. Mamat, and I. Mohd, The convergence properties of a new type of conjugate gradient methods, Appl. Math. Sci. 8 (2014), 33–44.
[19] M. Rivaie, M. Mamat, L.W. June, and I. Mohd, A new class of nonlinear conjugate gradient coefficients with global convergence properties, Appl. Math. Comput. 218 (2012), no. 22, 11323–11332.
[20] D. Touati-Ahmed and C. Storey, Efficient hybrid conjugate gradient techniques, J. Optim. Theory Appl. 64 (1990), no. 2, 379–397.
[21] X. Wang, and J. Chi, Cg global convergence properties with Goldstein line search, Bull. Brazil. Math. Soc. 36 (2005), no. 2, 197–204.
[22] Z. Wei and L. Li, and G. Qi, New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems, Appl. Math. Comput. 179 (2006), no. 2, 407–430.
[23] Z. Wei, S. Yao, and L. Liu, The convergence properties of some new conjugate gradient methods, Appl. Math. Comput. 183 (2006), no. 2, 1341–1350.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 11
November 2024
Pages 139-147
  • Receive Date: 25 September 2022
  • Accept Date: 27 October 2023