Journal of Applied MathematicsVolume 2012 (2012), Article ID 569795, 10 pagesdoi:10.1155/2012/569795

Research Article

A Mixed Spectral CD-DY Conjugate Gradient Method

Liu Jinkui,¹ Du Xianglin,¹ and Wang Kairong²

¹School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404100, China²College of Mathematics and Physics, Chongqing University, Chongqing 401331, China

Received 14 November 2011; Revised 22 January 2012; Accepted 23 January 2012

Academic Editor: Shan Zhao

Copyright © 2012 Liu Jinkui et al. 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.

A mixed spectral CD-DY conjugate descent method for solving unconstrained optimization problems is proposed, which combines the advantages of the spectral conjugate gradient method, the CD method, and the DY method. Under the Wolfe line search, the proposed method can generate a descent direction in each iteration, and the global convergence property can be also guaranteed. Numerical results show that the new method is efficient and stationary compared to the CD (Fletcher 1987) method, the DY (Dai and Yuan 1999) method, and the SFR (Du and Chen 2008) method; so it can be widely used in scientific computation.

1. Introduction

The purpose of this paper is to study the global convergence properties and practical computational performance of a mixed spectral CD-DY conjugate gradient method for unconstrained optimization without restarts, and with appropriate conditions.

Consider the following unconstrained optimization problem: where is continuously differentiable and its gradient is available. Generally, we use the iterative method to solve (1.1), and the iterative formula is given by where is the current iteration, is a positive scalar and called the step-size, which is determined by some line search, is the search direction defined by where and is a scalar which determines the different conjugate gradient methods [1, 2].

There are many kinds of iterative method that include the steepest descent method, Newton method, and conjugate gradient method. The conjugate direction method is a commonly used and effective method in optimization, and it only needs to use the information of the first derivative. However, it overcomes the shortcoming of the steepest descent method in the slow convergence and avoids the defects of Newton method in storage and computing the second derivative.

The original CD method was proposed by Fletcher [3], in which is defined by where denotes the Euclidean norm of vectors. An important property of the CD method is that the method in each iteration will produce a descent direction under the strong Wolfe line search: where . Dai and Yuan [4] first proposed the DY method, in which is defined by Dai and Yuan [4] also strictly proved that the DY method in each iteration produces a descent direction under the Wolfe line search (1.5) and Some good results about the CD method and DY method have also been reported in recent years [5–11].

Quite recently, Birgin and Martinez [12] proposed a spectral conjugate gradient method by combining conjugate gradient method and spectral gradient method. Unfortunately, the spectral conjugate gradient method [12] cannot guarantee to generate descent directions. So, based on the FR formula, Zhang et al. [13] modified the FR method such that the direction generated is always a descent direction. Based on the modified FR conjugate gradient method [13], Du and Chen [14] proposed a new spectral conjugate gradient method: where And they proved the global convergence of the modified spectral FR method (In this paper, we call it SFR method) with the mild conditions.

The observation of the above formula motivates us to construct a new formula; which combines the advantage of the spectral gradient method, CD method, and DY method as follows: where is specified by And under some mild conditions, we give the global convergence of the mixed spectral CD-DY conjugate gradient method with the Wolfe line search.

This paper is organized as follows. In Section 2, we propose the corresponding algorithm and give some assumptions and lemmas, which are usually used in the proof of the global convergence properties of nonlinear conjugate gradient methods. In Section 3, global convergence analysis is provided with suitable conditions. Preliminary numerical results are presented in Section 4.

2. Algorithm and Lemmas

In order to establish the global convergence of the proposed method, we need the following assumption on objective function, which have been used often in the literature to analyze the global convergence of nonlinear conjugate gradient methods with inexact line search.

Assumption 2.1. (i) The level set is bounded, where is the starting point.(ii) In some neighborhood of , the objective function is continuously differentiable, and its gradient is Lipchitz continuous, that is, there exists a constant such that Now we give the mixed spectral CD-DY conjugate gradient method as follows.

Algorithm 2.2. Step 1. Data , . Set ; if , then stop.Step 2. Compute by the strong Wolfe line search (1.5) and (1.8).Step 3. Let ,; if , then stop.Step 4. Compute by (1.12), and generate by (1.11).Step 5. Set ; go to Step 2.

The following lemma shows that Algorithm 2.2 produces a descent direction in each iteration with the Wolfe line search.

Lemma 2.3. Let the sequences and be generated by Algorithm 2.2, and let the step-size be determined by the Wolfe line search (1.5) and (1.8), then

Proof. The conclusion can be proved by induction. Since , the conclusion holds for . Now we assume that the conclusion is true for . Then from (1.8), we have If , then from (1.14) and (2.3), we have .From (1.4), (1.11), and (1.13), we have If , then from (1.14) and (2.3), we have .From (1.11), (1.7), and (1.13), we have From the above inequality (2.4) and (2.5), we obtain that the conclusion holds for .

The conclusion of the following lemma, often called the Zoutendijk condition, is used to prove the global convergence properties of nonlinear conjugate gradient methods. It was originally given by Zoutendijk [15].

Lemma 2.4 (see [15]). Suppose that Assumption 2.1 holds. Let the sequences and be generated by Algorithm 2.2, and let the step-size be determined by the Wolfe line search (1.5) and (1.8), and Lemma 2.3 holds. Then

Lemma 2.5. Let the sequences and be generated by Algorithm 2.2, and let the step-size be determined by the Wolfe line search (1.5) and (1.8), and Lemma 2.3 holds. Then

Proof. If , then from Lemma 2.3, we have . From (1.11), (1.4), and (1.13), we have From the above equation, we have If , from (2.5), we have From (1.12), (2.9), and (2.10), we obtain that the conclusion (2.7) holds.

3. Global Convergence Property

The following theorem proves the global convergence of the mixed spectral CD-DY conjugate gradient method with the Wolfe line search.

Theorem 3.1. Suppose that Assumption 2.1 holds. Let the sequences and be generated by Algorithm 2.2, and let the step-size be determined by the Wolfe line search (1.5) and (1.8), and Lemma 2.3 holds. Then

Proof. Suppose by contradiction that there exists a positive constant , such that holds for all .

From (1.11), we have , and by squaring it, we get From the above equation and Lemma 2.5, we have Dividing the above inequality by , we have Using (3.5) recursively and noting that , we get Then from (3.2) and (3.6), we have which indicates The above equation contradicts the conclusion of Lemma 2.4. Therefore, the conclusion (3.1) holds.

4. Numerical Experiments

In this section, we report some numerical results. We used MATLAB 7.0 to test some problems that are from [16] and compare the performance of the mixed spectral CD-DY method (Algorithm 2.2) with the CD method, DY method, and SFR method. The global convergence of the CD method has not still been proved under the Wolfe line search, so our line search subroutine computes such that the strong Wolfe line search conditions hold with and . We also use the condition or It- as the stopping criterion. It-max denotes the maximal number of iterations.

The numerical results of our tests are reported in the following table. The first column “Problem” represents the name of the tested problem in [16]. “Dim” denotes the dimension of the tested problem. The detailed numerical results are listed in the form NI/NF/NG, where NI, NF, and NG denote the number of iterations, function evaluations, and gradient evaluations respectively. If the limit of 9999 function evaluations was exceeded, the run was stopped; this is indicated by “—”.

In order to rank the average performance of all above methods, one can compute the total number of function and gradient evaluation by the formula where is some integer. According to the results on automatic differentiation [17, 18], the value of can be set to 5 That is to say, one gradient evaluation is equivalent to five function evaluations if automatic differentiation is used.

By making use of (4.2), we compare the mixed spectral CD-DY method as follows: for the th problem, compute the total number of function evaluations and gradient evaluations required by the CD method, the DY method, the SFR method, and the mixed spectral CD-DY method by formula (4.2), and denote them by , , , and ; then calculate the ratios:

From Table 1, we know that some problems are not run by some methods. So, if the th problem is not run by the given method, we use a constant instead of , where denotes the set of test problems, which can be run by the given method.

Table 1: The performance of the CD method, DY method, CD-DY method, and SFR method.

Problem
Dim
CD
DY
CD-DY
SFR

ROSE
2
88/250/223
64/190/172
60/188/168
64/190/172

FROTH
2
—
42/168/138
38/151/123
—

BADSCP
2
682/1885/1637
—
2704/6550/6438
—

BADSCB
2
272/941/800
—
726/2051/1786
—

BEALE
2
73/177/155
75/186/164
68/175/145
75/186/164

JENSAM

17/61/43
10/49/26
26/80/57
15/48/32

HELIX
3
56/157/132
37/118/98
50/145/120
37/118/98

BRAD
3
75/224/189
66/208/177
37/120/98
66/208/177

SING
4
454/1074/1009
2286/4555/4545
850/1894/1863
1476/2901/2891

WOOD
4
184/438/399
100/291/240
139/396/337
100/291/240

KOWOSB
4
173/516/449
536/1449/1271
144/421/365
504/1386/1211

BD
4
43/169/132
39/158/121
28/144/113
37/152/116

WATSON
5
89/279/239
127/348/299
158/438/373
128/352/304

BIGGS
6
200/579/509
294/824/712
236/680/599
288/812/703

OSB2
11
3243/5413/5398
7006/11059/11048
584/1262/1226
7013/11102/11091

VAEDIM
5
6/57/38
6/57/38
6/57/38
6/57/38

10
7/81/52
7/81/52
7/81/52
7/81/52

PEN1
50
2209/2565/2536
1727/2117/2043
116/221/190
1727/2117/2043

100
62/223/182
31/157/121
31/167/131
31/157/121

TRIG
100
—
305/399/398
88/145/144
305/399/398

500
—
343/424/423
109/189/188
344/427/425

ROSEX
500
92/267/238
68/207/186
65/205/182
68/207/186

1000
98/287/255
68/207/186
65/205/182
68/207/186

SINGX
100
682/1593/1517
1488/3139/3073
1159/2366/2614
2411/5326/5084

1000
511/1245/1135
2092/4451/4321
1374/3104/3064
5213/10042/10032

BV
500
1950/2543/2542
4796/6823/6822
1311/2131/2130
4784/6793/6792

1000
632/833/832
414/449/448
414/449/448
414/449/448

IE
500
7/15/8
7/15/8
7/15/8
7/15/8

1000
7/15/8
7/15/8
7/15/8
7/15/8

TRID
500
52/112/107
49/106/101
43/94/89
49/106/101

1000
70/149/145
64/137/133
55/119/115
64/137/133

The geometric mean of these ratios for the CD method, the DY method and SFR method, over all the test problems is defined by where denotes the set of the test problems and denotes the number of elements in . One advantage of the above rule is that, the comparison is relative and hence is not be dominated a few problems for which the method requires a great deal of function evaluations and gradient functions.

According to the above rule, it is clear that . From Table 2, we can see that average performance of the mixed spectral CD-DY conjugate gradient method (Algorithm 2.2) works the best. So, the mixed spectral CD-DY conjugate gradient method has some practical values.

Table 2: Relative efficiency of the CD, DY, SFR, and the mixed spectral CD-DY methods.

CD
DY
SFR
CD-DY

1.3956
1.6092
1.6580
1

Acknowledgments

The authors wish to express their heartfelt thanks to the referees and the editor for their detailed and helpful suggestions for revising the paper. This work was supported by The Nature Science Foundation of Chongqing Education Committee (KJ091104).

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