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A new type of quasi-newton updating formulas based on the new quasi-newton equation

  • * Corresponding author: Basim A. Hassan

    * Corresponding author: Basim A. Hassan
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  • The quasi-Newton equation is the very foundation of an assortment of the quasi-Newton methods. Therefore, by using the offered alternative equation, we derive the modified BFGS quasi-Newton updating formulas. In this paper, a new y-technique has been introduced to modify the secant equation of the quasi-Newton methods. Prove the global convergence of this algorithm is associated with a line search rule. The numerical results explain that the offered method is effectual for the known test problems.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Table 1.  Some modifications of QN-equations

    Author(s) QN conditions Ref.
    Powell $ B_{k+1}s_k=\tilde{y}_k= \varphi_k y_k +(1-\varphi_k)B_ks_k $ [8]
    Li and Fukushima $ B_{k+1}s_k=\tilde{y}_k= y_k +t_ks_k, t_k \le 10^{-6} $ [5]
    Wei, Li, and Qi $ B_{k+1}s_k=\tilde{y}_k= y_k +\frac{2(f_k-f_{k+1})+(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k $ [9]
    Zhang, Deng, and Chen $ B_{k+1}s_k=\tilde{y}_k= y_k +\frac{6(f_k-f_{k+1})+3(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k $ [18]
    Yuan and Wei $ B_{k+1}s_k=\tilde{y}_k= y_k +\frac{max(0, 2(f_k-f_{k+1})+(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k $ [14]
    Yuan, Wei and Wu $ B_{k+1}s_k=\tilde{y}_k= y_k +\frac{max(0, 6(f_k-f_{k+1})+3(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k $ [17]
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    Table 2.  Comparison of different BFGS-algorithms with different test functions and different dimensions

    P.No. n BFGS algorithm BBFGS with $ u_k=y_k $ BBFGS with $ u_k=g_{k+1} $
    NI NF NI NF NI NF
    1 2 35 140 36 124 8 29
    2 2 9 26 8 23 5 16
    3 2 43 166 34 123 3 12
    4 2 3 30 3 30 3 30
    5 2 15 50 15 48 5 17
    6 2 2 27 2 27 2 27
    7 3 34 113 26 86 7 20
    8 3 16 54 15 51 6 18
    9 3 2 4 2 4 2 4
    10 3 2 27 2 27 2 27
    11 3 2 27 2 27 2 27
    12 4 20 60 20 60 5 17
    13 4 19 61 24 73 4 13
    14 4 21 65 23 72 4 10
    15 4 17 54 16 49 5 17
    16 5 2 27 2 27 2 27
    17 6 25 72 33 101 4 12
    18 11 3 31 3 31 3 31
    19 20 31 102 33 103 4 13
    20 400 64 209 91 297 5 17
    21 400 2 27 2 27 2 27
    22 200 2 5 2 5 2 5
    23 100 2 27 2 27 2 27
    24 500 9 33 8 28 10 31
    25 500 2 4 2 4 2 4
    26 500 6 16 7 19 5 14
    27 500 57 281 16 114 5 17
    28 500 2 4 2 4 2 4
    29 500 3 7 3 7 3 7
    30 500 3 7 3 7 3 7
    Total 453 1756 437 1625 117 527
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    Table 3.  Relative efficiency of the new Algorithms

    BFGS algorithm BBFGS with $ u_k=y_k $ BBFGS with $ u_k=g_{k+1} $
    NI 100% 96.70% 25.82%
    NF 100% 92.53% 30.01%
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