Secant Condition Free of a Spectral Hestenses-Stiefel (SHS) Conjugate Gradient Method and its Sufficient Descent Properties

  • Authors

    • Usman Abbas Yakubu
    • Mustafa Mamat
    • Mohamad Afendee Mohamed
    • Puspa Liza Ghazali
    • Mohd Rivaie
    https://doi.org/10.14419/ijet.v7i3.28.23467
  • Global convergence, exact line search, spectral CG, secant condition, sufficient descent property.

  • The conjugate gradient method have been used widely to solve unconstrained minimization problems as a result of less storage locations and less computational expensive in dealing with the large-scale problems. In this work, we suggested a spectral HS conjugate gradient method without employing the secant condition and use some unconstrained problems with many variables to prove its sufficient descent as well as global convergence, the results is certified by apply exact line search procedure.

                                                                                                                                          

  • References

    1. [1] M. Raydan (1997). The Barzilai and J.M. Borwein gradient methods for the large scale unconstrained minimization in extreme problems. SIAM. J. Optim., 7 (1), 26-33.

      [2] E. Dolan and J. J. More (2002). Benchmarking optimization software with performance profile. Math. Prog., 91, 201-213.

      [3] E.G. Birgin and J. M. Martinez (2011). A spectral conjugate gradient method for unconstrained optimization. Appl. Math. Optim., 43 (2), 117-128.

      [4] X. Wu (2015). A new spectral Polak- Ribière -Polak conjugate gradient method. ScienceAsia, 41, 345-349.

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

      [6] J. C. Gilbert and J. Nocedal (1992). Global convergence properties of conjugate gradient methods for optimization. SIAM. J. Optim., 2, 21-42.

      [7] C. Hu and Z. Wan (2013). An extended spectral conjugate gradient method for unconstrained optimization problems. British Journal of Math. and Computer Science, 3, 86-98.

      [8] H. Huang, Z. Wei and Y. Shengwei (2007). The proof of the sufficient decent condition of the Wei-Yao-Liu conjugate gradient method under the strong Wolfe-Powell line search. Applied Mathematics and Computation, 189, 1241–1245.

      [9] M. J. D. Powell (1984). Non-convex minimization calculations and the conjugate gradient method. Lecture Notes in Mathematics, 1066, 122-241.

      [10] M. J. D. Powell (1977). Restart procedures for the conjugate gradient method. Math. Program., 12, 241-254.

      [11] J. Barzilai and J. M. Borwein (1988). Two-point step size gradient methods, IMA J Numer Anal, 8, 141-148.

      [12] X. Du and J. Liu (2011). Global convergence of a spectral HS conjugate gradient method. Procedia Engineering, 15, 1487-1492.

      [13] N. Zull, M. Rivaie, M. Mamat, Z Salleh and Z. Amani (2015). Global convergence of a spectral conjugate gradient by using strong Wolfe line search. Appl. Math. Sci., 63, 3105-3117.

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

      [15] N. Andrei (2008). An unconstrained optimization test functions collection, Adv. Modell. Optim., 10, 147-161.

      [16] A. Y. Usman, M. Mamat, M. Rivaie, A. M. Mohamad and B. Y. Rabi’u (2018). Secant free condition of a spectral WYL and its global convergence properties. Far East Journal of Mathematical Science, 103(12), 1889-1902.

      [17] A. Y. Usman, M. Mamat, M. Rivaie, A. M. Mohamad and J. Sabi’u (2018). A recent modification on Dai-Liao conjugate gradient method for solving symmetric nonlinear equations. Far East Journal of Mathematical Science, 103(12), 1961-1974.

      [18] K. U. Kamfa, M. Mamat, A. Abashar, M. Rivaie, P. L. B. Ghazali and Z. Salleh (2015). Another modified conjugate gradient coefficient with global convergence properties. Applied Mathematical Sciences, 9, 1833-1844.

      [19] N. Z. Abidin, M. Mamat, B. Dangerfield, J. H. Zulkepli, M. A. Baten and A. Wibowo (2014). Combating obesity through healthy eating behavior: A call for system dynamics optimization. Plos One, 9(12), 1-17.

      [20] M. Mamat, Y. Rokhayati, N. M. M. Noor and I. Mohd (2011). Optimizing human diet problem with puzzy price using fuzzy linear programming approach. Pakistan Journal of Nutrition, 10(6), 594-598.

      [21] A. Abashar, M. Mamat, M. Rivaie and I. Mohd (2014). Global convergence properties of a new class of conjugate gradient method for unconstrained optimization. Applied Mathematical Sciences Issue, 65-68, 3307-3319.

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

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  • How to Cite

    Abbas Yakubu, U., Mamat, M., Afendee Mohamed, M., Liza Ghazali, P., & Rivaie, M. (2018). Secant Condition Free of a Spectral Hestenses-Stiefel (SHS) Conjugate Gradient Method and its Sufficient Descent Properties. International Journal of Engineering & Technology, 7(3.28), 312-315. https://doi.org/10.14419/ijet.v7i3.28.23467