Solving a large scale nonlinear unconstrained optimization problems by using new coefficient of conjugate gradient method with exact line search direction

 
 
 
  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract


    In this paper, an efficient modification of nonlinear conjugate gradient method and an associated implementation, based on an exact line search, are proposed and analyzed to solve large-scale unconstrained optimization problems. The method satisfies the sufficient descent property. Furthermore, global convergence result is proved. Computational results for a set of unconstrained optimization test problems, some of them from CUTE library, showed that this new conjugate gradient algorithm seems to converge more stable and outperforms the other similar methods in many situations.

     

     

     

  • Keywords


    Conjugate gradient; Exact line search; Global convergence; Sufficient descent condition; Unconstrained optimization.

  • References


      [1] Stiefel, E. "Methods of conjugate gradient for solving linear equation." J. Res. Nat. Bur. Standards 49 (1952): 409-436.

      [2] Fletcher, Reeves, and Colin M. Reeves. "Function minimization by conjugate gradients." Computer Journal 7(2), (1964): 149-154.

      [3] Polyak, Boris T. "The conjugate gradient method in extremal problems." USSR Computational Mathematics and Mathematical Physics 9(4), (1969): 94-112.

      [4] Polak, Elijah, and Gerard Ribiere. "Note sur la convergence de méthodes de directions conjuguées." Revue française d'informatique et de recherche opérationnelle. Série rouge 3(16), (1969): 35-43.

      [5] R.Fletcher, practical method of optimization, Unconstrained optimization, Wiley, New York, (1997): 214-242.

      [6] Liu, Y., and C. Storey. "Efficient generalized conjugate gradient algorithms, part 1: Theory." Journal of Optimization Theory and Applications 69(1) (1991): 129-137.

      [7] Dai, Yu-Hong, and Yaxiang Yuan. "A nonlinear conjugate gradient method with a strong global convergence property." SIAM Journal on Optimization 10, 1 (1999): 177-182

      [8] Zoutendijk, G. "Nonlinear programming, computational methods." Integer and Nonlinear Programming (1970): 37-86.

      [9] Powell, Michael James David. "Restart procedures for the conjugate gradient method." Mathematical Programming 12, 1 (1977): 241-254.

      [10] Powell, Michael JD. "Nonconvex minimization calculations and the conjugate gradient method." In Numerical Analysis, pp. Springer, Berlin, Heidelberg, (1984). 122-141.

      [11] Powell, M. JD. "Convergence properties of algorithms for nonlinear optimization." Siam Review 28(4), (1986): 487-500.

      [12] Wei, Zengxin, Guoyin Li, and Liqun Qi. "New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems." Applied Mathematics and Computation 179(2), (2006): 407-430.

      [13] Dai, Zhi-Feng. "Two modified HS type conjugate gradient methods for unconstrained optimization problems." Nonlinear Analysis: Theory, Methods and Applications 74(3) (2011): 927-936.

      [14] Rivaie, Mohd, Mustafa Mamat, Leong Wah June, and Ismail Mohd. "A new class of nonlinear conjugate gradient coefficients with global convergence properties." Applied Mathematics and Computation 218(22), (2012): 11323-11332.

      [15] Aini, N., Rivaie, M., and Mamat, M. "A modified conjugate gradient coefficient with inexact line search for unconstrained optimization". AIP Conference Proceedings, 1739(1), 020082, 2016.

      [16] Kamfa, Kamilu Uba, Mustafa Mamat, Abdelrhaman Abashar, Mohd Rivaie, Puspa Liza Binti Ghazali, and Zabidin Salleh. "Another modified conjugate gradient coefficient with global convergence properties." (2017). 9(37): 1833–1844.

      [17] Shapiee, Norrlaili, Mohd Rivaie, and Mustafa Mamat. "A new classical conjugate gradient coefficient with exact line search." AIP Conference Proceedings, 1739(1), 020082, 2016.

      [18] Hamoda, M., M. Rivaie, M. Mamat, and Z. Salleh. "A new nonlinear conjugate gradient coefficient for unconstrained optimization." Applied Mathematical Sciences 9 (2015): 1813-1822.

      [19] Al-Baali, Mehiddin. "Descent property and global convergence of the Fletcher—Reeves method with inexact line searches." IMA Journal of Numerical Analysis 5(1), (1985): 121-124

      [20] Li, Min, and Heying Feng. "A sufficient descent LS conjugate gradient method for unconstrained optimization problems." Applied Mathematics and Computation 218(5), (2011): 1577-1586.

      [21] N. Andrei, "40 conjugate gradients algorithms for unconstrained optimization", Bull. Malay. Math. Sci. Soc. 34 (2011): 319–330.

      [22] Wei, Zengxin, Shengwei Yao, and Liying Liu. "The convergence properties of some new conjugate gradient methods." Applied Mathematics and computation 183(2), (2006): 1341-1350.

      [23] Zhang, Yueqin, Hao Zheng, and Chuanlin Zhang. "Global convergence of a modified PRP conjugate gradient method." Procedia Engineering 31 (2012): 986-995.

      [24] Dai, Zhifeng, and Fenghua Wen. "Another improved Wei–Yao–Liu nonlinear conjugate gradient method with sufficient descent property." Applied Mathematics and Computation 218, 14 (2012): 7421-7430.

      [25] Dai, Yu-Hong, and Yaxiang Yuan. "Nonlinear conjugate gradient methods." Shanghai Science and Technology Publisher, Shanghai (2000).

      [26] Yuan, Yaxiang, and Wenyu Sun. "Theory and methods of optimization." (1999).

      [27] Dolan, Elizabeth D., and Jorge J. Moré. "Benchmarking optimization software with performance profiles." Mathematical Programming 91(2), (2002):201-213.

      [28] Kenneth E. "A simulation test approach to the evaluation of nonlinear optimization algorithms." ACM Transactions on Mathematical Software, 3(4), (1977): 305-315.

      [29] N. Andrei, "An unconstrained optimization test functions collection," Advanced Modeling and Optimization, 10 (2008), 147-161.

      [30] Mamat, M., Deraman, S.K., Noor, N.M.M., Rokhayati, Y. Diet problem and nutrient requirement using fuzzy linear programming approach (2012) Asian Journal of Applied Sciences, 5(1): 52-59.

      [31] Abidin, N.Z., Mamat, M., Ganderfield, B., Zulkefli, J.H., Baten, Md. A., Wibowo, A. Combating obesity through healthy eating behavior: A call for system dynamics optimization, Plos One, 9(12), e114135.


 

View

Download

Article ID: 20976
 
DOI: 10.14419/ijet.v7i3.28.20976




Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.