Abstract and Applied Analysis

Volume 2013 (2013), Article ID 742815, 5 pages

http://dx.doi.org/10.1155/2013/742815

Research Article

Nonlinear Conjugate Gradient Methods with Wolfe Type Line Search

Yuan-Yuan Chen and Shou-Qiang Du

College of Mathematics, Qingdao University, Qingdao 266071, China

Received 20 January 2013; Accepted 6 February 2013

Academic Editor: Yisheng Song

Copyright © 2013 Yuan-Yuan Chen and Shou-Qiang Du. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Nonlinear conjugate gradient method is one of the useful methods for unconstrained optimization problems. In this paper, we consider three kinds of nonlinear conjugate gradient methods with Wolfe type line search for unstrained optimization problems. Under some mild assumptions, the global convergence results of the given methods are proposed. The numerical results show that the nonlinear conjugate gradient methods with Wolfe type line search are efficient for some unconstrained optimization problems.

1. Introduction

In this paper, we focus our attention on the global convergence of nonlinear conjugate gradient method with Wolfe type line search. We consider the following unconstrained optimization problem:

In (1), is continuously differentiable function, and its gradient is denoted by . Of course, the iterative methods are often used for (1). The iterative formula is given by

where ,   is the kth and (k+1)th iterative step, is a step size, and is a search direction. Here, in the following, we define the search direction by

In (3), is a conjugate gradient scalar, and the well-known useful formulas are , , , and (see [1–6]). Recently, some kinds of new nonlinear conjugate gradient methods are given in [7–11]. Based on the new method, we give some new kinds of nonlinear conjugate gradient methods and analyze the global convergence of the methods with Wolfe type line search.

The rest of the paper is organized as follows. In Section 2, we give the methods and the global convergence results for them. In the last section, numerical results and some discussions are given.

2. The Methods and Their Global Convergence Results

Firstly, we give the Wolfe type line search, which will be used in our new nonlinear conjugate gradient methods. In the following section of this paper, stands for the 2-norm.

We have used the Wolfe type line search in [12].

The line search is to compute such that

where .

Now, we present the nonlinear conjugate gradient methods as follows.

Algorithm 1. We have the following steps.

Step 0. Given , set . If , then stop.

Step 1. Find satisfying (4) and (5), and by (2), is given. If , then stop.

Step 2. Compute by the following equation:

in which . Set , and go to Step 1.

Before giving the global convergence theorem, we need the following assumptions.

Assumption 1. (A1) The set is bounded.

(A2) In the neighborhood of , denoted as , is continuously differentiable. Its gradient is Lipschitz continuous; namely, for , there exists such that

In order to establish the global convergence of Algorithm 1, we also need the following lemmas.

Lemma 2. Suppose that Assumption 1 holds; then, (4) and (5) are well defined.

The proof is essentially the same as Lemma 1 of [12]; hence, we do not rewrite it again.

Lemma 3. Suppose that direction is given by (6); then, one has

holds for all . So, one knows that is descent search direction.

Proof. From the definitions of and , we can get it.

Lemma 4. Suppose that Assumption 1 holds, and is determined by (4) and (5); one has

Proof. By (4), (5), Lemma 3, and Assumption 1, we can get

Then, we know that

By squaring both sides of the previous inequation, we get

By (4), we know that

So, we get (9), and this completes the proof of the lemma.

Lemma 5. Suppose that Assumption 1 holds, is computed by (6), and is determined by (4) and (5); one has

Proof. From Lemmas 3 and 4, we can obtain (14).

Theorem 6. Consider Algorithm 1, and suppose that Assumption 1 holds. Then, one has

Proof. We suppose that the theorem is not true. Suppose by contradiction that there exists such that

holds for .

From (6) and Lemma 3, we get

Dividing the previous inequation by , we get

So, we obtain

which contradicts (14). Hence we get this theorem.

Remark 7. In Algorithm 1, we also can use the following equations to compute :

where ;

where .

Algorithm 8. We have the following steps.

Step 0. Given , set , . If , then stop.

Step 1. Find satisfying (4) and (5), and by (2), is given. If , then stop.

Step 2. Compute by formula

and compute by (3). Set , and go to Step 1.

Lemma 9. Suppose that Assumption 1  holds, and is computed by (22); if , then one gets for all and (see [9]).

Lemma 10. Suppose that , and is a constant. If positive series satisfied

one has

From the previous analysis, we can get the following global convergence result for Algorithm 8.

Theorem 11. Suppose that Assumption 1 holds, and , where is a constant. Then, one has

Proof. Suppose by contradiction that there exists such that

holds for all . From (3), we have

Squaring both sides of the previous equation, we get

Let and ; from (22), we have

By ,  , we know that

By (30), we get

From (31) and Lemma 10, we have

which contradicts Lemma 4. Therefore, we get this theorem.

Algorithm 12. We have the following steps.

Step 0. Given , , set . If , then stop.

Step 1. Find satisfying (4) and (5), and by (2), is given. If , then stop.

Step 2. Compute by

where

Set , and go to Step 1.

Lemma 13. Suppose that direction is given by (33) and (34); then, one has

holds for any .

Lemma 14. Suppose that Assumption 1 holds, is generated by (33) and (34), and is determined by (4) and (5); one has

Proof. From Lemma 4 and Lemma 13, we obtain (36).

Lemma 15. Suppose that is convex. That is, , for all , where is the Hessian matrix of . Let and be generated by Algorithm 12; one has

Proof. By Taylor’s theorem, we can get

where , and .

By Assumption 1, (4), and (38), we get

So, we get (37).

Theorem 16. Consider Algorithm 12, and suppose that Assumption 1 and the assumption of Lemma 15 hold. Then, one has

Proof. We suppose that the conclusion is not true. Suppose by contradiction that there exists such that

holds for all .

By Lemma 13, we have

From Assumption 1, Lemma 15, and (42), we know that

Therefore, by (33), we get

We obtain

which contradicts (36). Therefore, we have

So, we complete the proof of this theorem.

Remark 17. In Algorithm 12, can also be computed by the following formula:

where .

3. Numerical Experiments and Discussions

In this section, we give some numerical experiments for the previous new nonlinear conjugate gradient methods with Wolfe type line search and some discussions. The problems that we tested are from [13]. We use the condition as the stopping criterion. We use MATLAB 7.0 to test the chosen problems. We give the numerical results of Algorithms 1 and 12 to show that the method is efficient for unconstrained optimization problems. The numerical results of Algorithms 1 and 12 are listed in Tables 1 and 2.

Table 1: Test results for Algorithm 1.

Name
Dim
NI
NF
NG

GULF
3
2
52
3

VARDIM
2
3
54
6

LIN
50
1
3
3

LIN
2
1
3
3

LIN
1000
1
3
3

LIN1
2
1
51
2

LIN1
10
1
3
3

LIN0
4
1
3
3

Name: the test problem name; Dim: the problem dimension; NI: the iterations number; NF: the function evaluations number; NG: the gradient evaluations number.

Table 2: Test results for Algorithm 12.

Name
Dim
NI
NF
NG

GULF
3
2
52
3

BIGGS
6
1000
1149
1095

IE
50
1000
1053
1003

IE
3
1000
1101
1052

TRIG
1000
1103
1052
3

BV
10
5
203
55

BV
3
6
109
9

Name: the test problem name; Dim: the problem dimension; NI: the iterations number; NF: the function evaluations number; NG: the gradient evaluations number.

Discussion 1. From the analysis of the global convergence of Algorithm 1, we can see that if satisfies the property of efficient descent search direction, we can get the global convergence of the corresponding nonlinear conjugate gradient method with Wolfe type line search without other assumptions.

Discussion 2. In Algorithm 8, we use a Wolfe type line search. Overall, we also feel that nonmonotone line search (see [14]) also can be used in our algorithms.

Discussion 3. From the analysis of the global convergence of Algorithm 12, we can see that when is an efficient descent search direction, we can get the global convergence of the corresponding conjugate gradient method with Wolfe type line search without requiring uniformly convex function.

Acknowledgments

This work is supported by the National Science Foundation of China (11101231 and 10971118), Project of Shandong Province Higher Educational Science and Technology Program (J10LA05), and the International Cooperation Program for Excellent Lecturers of 2011 by Shandong Provincial Education Department.

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