We propose and generalize a new nonlinear conjugate gradient method for unconstrained optimization. The global convergence is proved with the Wolfe line search. Numerical experiments are reported which support the theoretical analyses and show the presented methods outperforming CGDESCENT method.
This paper is concerned with conjugate gradient methods for unconstrained optimization
Different conjugate gradient algorithms correspond to different choices for the scale parameter
These motivate us to derive some efficient algorithms. In this paper, we focus on mixed conjugate gradient methods. These methods are combinations of different conjugate gradient methods. The aim of this paper is to propose the new methods that possess both convergence and well numerical results.
The line search in the conjugate gradient algorithms is often based on the Wolfe inexact line search
Many research on the parameter
PRP is famous as the best performance of all conjugate gradient methods which is the restart method in nature. When the direction
HS is similar to PRP. It is equal to PRP when using the precision line search. HS satisfies the conjugate condition which is different from other methods.
TouatiAhmed and Storey [
Hager and Zhang (CGDESCENT) [
Zhang et al. (ZZL) [
Consider the above mixed techniques and the properties of the classical conjugate gradient methods, the new mixed methods will be presented. The main difference between the new methods and the existed methods are the choice of
Firstly, we present a new formula
The rest of the paper is organized as follows. In Section
We discuss a new mixed conjugate gradient method
In order to derive the global convergence of the algorithm, we use the following assumptions.
The objective function
Suppose that H 2.1 and H 2.2 hold. If the conjugate gradient method satisfies
Suppose that H 2.1 and H 2.2 hold. Let
The conclusion can be proved by induction. When
When
When
Thus, the theorem is proved.
Suppose that H 2.1 and H 2.2 hold. Consider Algorithm
By contradiction, assume that (
From (
By (
Then,
So,
From (
By squaring the two sides of (
The generalization of the new mixed method is as follows:
Suppose that H 2.1 and H 2.2 hold. Let
The conclusion can be proved by induction. When
When
To sum up, the theorem is proved.
Suppose that H 2.1 and H 2.2 hold. Consider Algorithm
By contradiction, assume that (
From (
By (
Then,
This section is devoted to test the implementation of the new methods. We compare the performance of the new methods with the CGDESCENT and ZZL methods.
All tests in this paper are implemented on a PC with 1.8 MHz Pentium IV and 256 MB SDRAM using MATLAB 6.5. If
In the table, the four reported data
Numerical results of the NEW1 and NEW2.
Number  Prob.  dim 




Powell badly scaled 




 

Brown badly scaled 







Trigonometric function 




 

Chebyquad 




 

Penalty function 














Allgower 




 

Variable dimension 














Penalty function 







Integral equation 




 

Separable cubic 












Numerical results of the CGDESCENT and NEW3.
Number  Prob.  dim 




Powell badly scaled 




 

Brown badly scaled 




 

Trigonometric function 




 

Chebyquad 







Penalty function 














Allgower 




 

Variable dimension 














Penalty function 







Integral equation 




 

Separable cubic 












Numerical results of the NEW4 and NEW5.
Number  Prob.  dim 




Powell badly scaled 




 

Brown badly scaled 




 

Trigonometric function 




 

Chebyquad 




 

Penalty function 














Allgower 




 

Variable dimension 














Penalty function 




 

Integral equation 




 

Separable cubic 












Numerical results of the NEW6 and NEW7.
Number  Prob.  dim 




Powell badly scaled 




 

Brown badly scaled 







Trigonometric function 




 

Chebyquad 




 

Penalty function 














Allgower 




 

Variable dimension 














Penalty function 




 

Integral equation 




 

Separable cubic 












Compared with the CGDESCENT method, the new methods are effective (see Table
















Using the formula
We use
It is obvious that
This work is supported in part by the NNSF (11171003) of China, Key Project of Chinese Ministry of Education (no. 211039), and Natural Science Foundation of Jilin Province of China (no. 201215102).