# American Institute of Mathematical Sciences

April  2017, 13(2): 595-608. doi: 10.3934/jimo.2016034

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

 1 Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu Province, 211106, China 2 Huaibei Normal University, Huaibei, Anhui Province, 235000, China 3 Hubei Engineering University, Xiaogan, Hubei Pvovince, 432000, China

Received  January 2015 Revised  December 2015 Published  May 2016

Fund Project: 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).

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.

Citation: Guanghui Zhou, Qin Ni, Meilan Zeng. A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 595-608. doi: 10.3934/jimo.2016034
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##### References:
CPU time performance profiles in a log2 scale (n = 102)
Iterations performance profiles in a log2 scale (n = 102)
CPU time performance profiles in a log2 scale (n = 103)
Iterations performance profiles in a log2 scale (n = 103)
CPU time performance profiles in a log2 scale (n = 104)
Iterations performance profiles in a log2 scale (n = 104)
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
 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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