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

Guanghui Zhou; Qin Ni; Meilan Zeng

doi:10.3934/jimo.2016034

Article Contents

2017, Volume 13, Issue 2: 595-608. Doi: 10.3934/jimo.2016034

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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: August 2017

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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Abstract

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.

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

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

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

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

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

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

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

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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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