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A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems

The authors are supported by the Natural Science Foundation of China (11471159,11571169), the Natural Science Foundation of Jiangsu Province (BK20141409), the Education Department Foundation of Anhui Province(KJ2016A651,2014jyxm161), and the Science and Technology Foundation of the Department of Education of Hubei Province (D20152701).
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  • In this paper, a scaled method that combines the conjugate gradient with moving asymptotes is presented for solving the large-scaled nonlinear unconstrained optimization problem. A diagonal matrix is obtained by the moving asymptote technique, and a scaled gradient is determined by multiplying the gradient with the diagonal matrix. The search direction is either a scaled conjugate gradient direction or a negative scaled gradient direction under different conditions. This direction is sufficient descent if the step size satisfies the strong Wolfe condition. A global convergence analysis of this method is also provided. The numerical results show that the scaled method is efficient for solving some large-scaled nonlinear problems.

    Mathematics Subject Classification: Primary: 65K05; Secondary: 90C30, 49M37.

    Citation:

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  • Figure 1.  CPU time performance profiles in a log2 scale (n = 102)

    Figure 2.  Iterations performance profiles in a log2 scale (n = 102)

    Figure 3.  CPU time performance profiles in a log2 scale (n = 103)

    Figure 4.  Iterations performance profiles in a log2 scale (n = 103)

    Figure 5.  CPU time performance profiles in a log2 scale (n = 104)

    Figure 6.  Iterations performance profiles in a log2 scale (n = 104)

    Table 1.  Test functions

    No. Function Name No. Function Name
    1 ARWHEAD 17 Ext quadratic penalty QP2
    2 COSINE 18 Ext quadratic exponential EP1
    3 EDENSCH 19 Ext Tridiagonal 2
    4 EG2 20 Ext DENSCHNF
    5 ENGVAL1 21 HIMMELBG
    6 GENROSE 22 HIMMELH
    7 Ext Freudenstein & Roth 23 TOINTGSS
    8 Raydan 2 24 Extended DENSCHNB
    9 Ext Tridiagonal 1 25 LIARWHD
    10 Ext TET 26 Extended Trigonometric
    11 Diagonal 5 27 Extended Penalty
    12 Diagonal 2 28 Extended BD1
    13 Ext Maratos 29 Perturbed Quadratic
    14 Ext Cliff 30 Raydan 1
    15 Perturbed quadratic diagonal 31 Diagonal 4
    16 Ext quadratic penalty QP1 32 QUARTC
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    [13] Y. NarushimaH. Yabe and J. A. Ford, A three-term conjugate gradient method with sufficient descent property for unconstrained optimization, SIAM Journal on Optimization, 21 (2011), 212-230.  doi: 10.1137/080743573.
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