An Extension of Polak-Ribière-Polyak Method

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    The conjugate gradient method has been widely used for finding solution for the large-scale unconstrained optimization. Fields such as computer science and engineering are the two most frequently engaged, because of its simplicity, the speed of getting the solution and the minimal storage requirement. This study presents an extended conjugate gradient method of Polak-Ribière-Polyak with the strong Wolfe-Powell (SWP) line search satisfying some properties such as sufficient descent and global convergence. For the purpose of experimentation, a set of 141 test problems have been used. The results showed that our proposed method has surpass the others in terms of efficiency and robustness.

                                                                                                                                          


  • Keywords


    Conjugate gradient method; global convergence; strong Wolfe-Powell; sufficient descent property; unconstrained optimization.

  • References


      [1] Cai J., Li Q., Li L., Peng H., & Yang Y., “A fuzzy adaptive chaoticant swarm optimization for economic dispatch”, International Journal of Electrical Power and Energy Systems, Vol. 34, No. 1, (2012), pp. 154–160.

      [2] Hestenes M. R. & Steifel E., “Method of conjugate gradient for solv-ing linear equations”, J. Res. Nat. Bur. Stand., 49, (1952), 409-436.

      [3] Fletcher R. & Reeves C., “Function minimization by conjugate gradients”, Computer Journal, Vol. 7, No. 2, (1964), pp. 149-154.

      [4] Polak E. & Ribiere G., Note sur la convergence de directions con-jugees. Rev. Francaise Informat Recherche Operationelle, 3E Annee 16, (1969), 35-43.

      [5] Gilbert J. C. & Nocedal J., “Global convergence properties of conjugate gradient methods for optimization”, SIAM Journal on optimization, Vol. 2, No. 1, (1992), pp. 21-42.

      [6] Fletcher R., Practical methods of optimization, John Wiley and Sons, 2001.

      [7] Liu Y. & Storey C., “Efficient generalized conjugate gradient al-gorithms, part 1: theory”, Journal of Optimization Theory and Ap-plications, Vol. 69, No. 1, (1991), pp. 129-137.

      [8] Dai Y. H. & Yuan Y., Nonlinear conjugate gradient methods, Shanghai Science and Technology Publisher, 2000.

      [9] Wei Z., Yao S., & Liu L., “The convergence properties of some new conjugate gradient methods”, Applied Mathematics and Computation, Vol. 183, No. 2, (2006), pp. 1341–1350.

      [10] Zoutendjik G., Nonlinear programming computational methods, in J. Abadie (Ed.), Interger and Nonlinear Programming, Amsterdam: North-Holland, (1970), pp. 37-86.

      [11] Al-Baali M., “Descent property and global convergence of the Fletcher–Reeves method with inexact line search”, IMA Journal of Numerical Analysis, Vol. 5, No. 1, (1985), pp. 121-124.

      [12] Guanghui L., Jiye H., & Hongxia Y., “Global convergence of the fletcher-reeves algorithm with inexact line search”, Applied Mathematics-A Journal of Chinese Universities, Vol. 10, No. 1, (1995), pp. 75–82.

      [13] Powell M. J. D., “Nonconvex minimization calculations and the conjugate gradient method”, Lecture Notes in Mathematics, 1066, (1984), pp. 122-141.

      [14] Shengwei Y., Wei Z., & Huang H., “A note about WYL’s conjugate gradient method and its applications”, Applied Mathematics and Computation, Vol. 191, No. 2, (2007), pp. 381–388.

      [15] Dai Z. & Wen F., “Another improved Wei–Yao–Liu nonlinear conjugate gradient method with sufficient descent property”, Applied Mathematics and Computation, Vol. 218, No. 14, (2012), pp. 7421–7430.

      [16] Zhang L., “An improved Wei-Yao-Liu nonlinear conjugate gradient method for optimization computation”, Applied Mathematics and Computation, Vol. 215, No. 6, (2009), pp. 2269–2274.

      [17] Rivaie M., Mamat M., June L. W., & Mohd I., “A new class of nonlinear conjugate gradient coefficients with global convergence properties”, Applied Mathematics and Computation, Vol. 218, No. 22, (2012), pp. 11323–11332.

      [18] Alhawarat A., Mamat M., Rivaie M., & Mohd I., “A new modification of nonlinear conjugate gradient coefficients with global convergence properties”, International Journal of Mathematical, Computational, Statistical, Natural and Physical Engineering, Vol. 8, No. 1, (2014), pp. 54-60.

      [19] Rivaie M., Mamat M., & Abashar A., “A new class of nonlinear conjugate gradient coefficients with exact and inexact line searches”, Applied Mathematics and Computation, Vol. 268, (2015), pp. 1152-1163.

      [20] Abashar A., Mamat M., Rivaie M., Mohd I., & Omer O., “The proof of sufficient descent condition for a new type of conjugate gradient methods”, AIP Conference Proceedings, Vol. 1602, (2014), pp. 296-303.

      [21] Bongartz I., Conn A. R., Gould N., & Toint P. L., “CUTE: constrained and unconstrained testing environment”, ACM Transactions on Mathematical Software, Vol. 21, No. 1, (1995), pp. 123–160.

      [22] Andrei N., “An unconstrained optimization test functions collec-tion”, Adv. Model. Optim, Vol. 10, No. 1, (2008), pp. 147-161.

      [23] Adorio E. P. & Diliman U. P., “Mvf-multivariate test functions library in C for unconstrained global optimization”, 2005, http://www.geocities.ws/eadorio/mvf.pdf.

      [24] Dolan E. D. & Jorge J. M., “Benchmarking optimization software with performance profiles”, Mathematical Programming, Vol. 91, No. 2, (2002), pp. 201-213.

      [25] Abidin N. Z., Mamat M., Dangerfield B., Zulkepli J. H., Baten M. A., & Wibowo A., “Combating obesity through healthy eating behavior: a call for system dynamics optimization”, Plos One, Vol. 9, No. 12, (2014), pp. 1-17.

      [26] Mamat M., Subiyanto, & Kartono A., “Mathematical model of cancer treatments using immunotherapy, chemotherapy and biochemotherapy”, Appl. Math. Sci, Vol. 7, No. (5-8), (2013), 247-261.

      [27] Mamat M., Deraman S. K., Noor N. M. M., & Rokhayati Y., “Diet problem and nutrient requirement using fuzzy linear programming approach”, Asian Journal of Applied Sciences, Vol. 5, No. 1, (2012), pp. 52-59.

      [28] Rivaie M., Mamat M., Mohd I., & Fauzi M., “Comparative study of conjugate gradient coefficient for unconstrained optimization”, Aus. J. Bas. Appl. Sci, Vol. 5, No. 9, (2011), pp. 947-951.

      [29] Mamat M., Rokhayati Y., Noor N. M. M., & Mohd I., “Optimizing human diet problem with fuzzy price using fuzzy linear programming approach”, Pakistan Journal of Nutrition, Vol. 10, No. 6, (2011), pp. 594-598.


 

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Article ID: 27383
 
DOI: 10.14419/ijet.v7i3.28.27383




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