New Hybrid Conjugate Gradient Method with Global Convergence Properties under Exact Line Search

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    Conjugate Gradient (CG) method is a very useful technique for solving large-scale nonlinear optimization problems. In this paper, we propose a new formula for 12خ²k"> , which is a hybrid of PRP and WYL methods. This method possesses sufficient descent and global convergence properties when used with exact line search. Numerical results indicate that the new formula has higher efficiency compared with other classical CG methods.

     


  • Keywords


    Nonlinear optimization; conjugate gradient coefficient; exact line search; global convergence; large scale.

  • References


      [1] M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49(1952), 409-436.

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

      [3] E. Polak, and G. Ribiere, Note Sur la convergence de directions conjuge`es, ESAIM: Mathematical Modelling and Numerical Analysis, 3E(1969), 35-43.

      [4] B. T. Polyak, The conjugate gradient method in extreme problems, USSRComputational Mathematics and Mathematical Physics, 9(1969), 94-112.

      [5] R. Fletcher, Practical Method of Optimization, New York, 2000.

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

      [7] Y. H. Dai and Y. X. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM Journal on Optimization, 10(1999), 177-182.

      [8] M. Mamat, M. Rivaie, I. Mohd and M. Fauzi (2010). A new conjugate gradient coefficient for unconstrained optimization, Int. J. Contemp. Math. Sciences, 5(29), 1429-1437.

      [9] M. Hamoda, A. Abashar, M. Mamat and M. Rivaie, A comparative study of two new conjugate gradient methods, AIP Conference Proceedings, 1643(2015), 616-621.

      [10] M. Rivaie, A. Abashar, M. Mamat and I. Mohd (2014). The convergence properties of a new type of conjugate gradient methods, 8(1), 33-44.

      [11] Aini, N., Hajar, N. Mamat, M., Zull, N., and Rivaie, M. (2017). Hybrid quasi-newton andconjugate gradient method for solving unconstrained optimization problems, Journal of Engineering and Applied Sciences, 12(8), 4627-4631.

      [12] Ghani, N. H. A., Mamat, M., Susilawati, F., Zull, N., Mohamed, N. S., and Reivaie, M. (2017). Another descent conjugate gradient methodwith strong-Wolfe line search. Journal of Engineering and Applied Sciences, 12(18), 4632-4636.

      [13] Ghani, N. H. A., Mohamed, N. S. Zull, N., Shoid, S., Rivaie, M., and Mamat (2017). Performance comparison of a new hybrid conjugate gradient method under exact and inexact line searches. Journal of Physics: Conference Series, 890(1).

      [14] Ghani, N. H. A., Rivaie, M., Mamat, M. A modified form of conjugate gradient method for unconstrained optimization problems. (2016), AIP Conference Proceedings, 1739, 020076.

      [15] Hager, W. W. and H. Zhang, A survey of nonlinear conjugate gradient methods. Pacific Journal of Optimization, 2006, 2(1). 35-58.

      [16] Mohamed, N. S., Mamat, M., and Rivaie, M. (2017). A new nonlinear conjugate gradient coefficient under strong Wolfe-Powell line search. AIP Conference Proceedings, 1870

      [17] Wan Osman, W. F. H., Hery Ibrahim, M. A., and Mamat, M. (2017). Hybrid DFP-CG method for solving unconstrained optimization problems. Journal of Physics: Conference Series, 890(1).

      [18] Zull, N., Aini, N., Mamat, M., Ghani, N. H. A., Mohamed, N. S., Rivaie, M. and Sukono. (2017). An alternative approach for conjugate gradient method. Journal of Engineering and Applied Sciences, 12(18), 4622-4626.

      [19] I. S. Mohammed, M. Mamat, A. Abashar, M. Rivaie and Z. Salleh (2017). The Global Convergence Properties of an Improved Conjugate Gradient Method. Journal of Applied Mathematical Sciences, 9(38), 1857-1868.

      [20] G. Zoutendijk, Nonlinear Programming, Computational Methods, Chapter in Integer and Nonlinear Programming, North-Holland, Amsterdam, 1970, 37-86.

      [21] Z.-X. Wei, S. W. Yao and L. Y. Liu, The convergence properties of some new conjugate gradient methods, Applied Mathematics and Computation, 183(2006), 1341-1350.

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

      [23] M. Hamoda, M. Mamat, M .Rivaie and Z. Salleh (2016). A Conjugate Gradient Method with Strong Wolfe-Powell Line Search for Unconstrained Optimization. Journal of Applied Mathematical Sciences, 10(15), 721-734.

      [24] K. E. Hillstrom, A simulation test approach to the evaluation of nonlinear optimization algorithm, ACM Transactions on Mathematical Software, 3(4) (1977), 305-315.

      [25] E. D. Dolan and J. J. Mor, Benchmarking optimization software with performance profiles, Mathematical Programming, 91(2002), 201-213.


 

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Article ID: 20965
 
DOI: 10.14419/ijet.v7i3.28.20965




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