Open Access

A modified three-term PRP conjugate gradient algorithm for optimization models

Journal of Inequalities and Applications20172017:97

https://doi.org/10.1186/s13660-017-1373-4

Received: 25 March 2017

Accepted: 20 April 2017

Published: 3 May 2017

Abstract

The nonlinear conjugate gradient (CG) algorithm is a very effective method for optimization, especially for large-scale problems, because of its low memory requirement and simplicity. Zhang et al. (IMA J. Numer. Anal. 26:629-649, 2006) firstly propose a three-term CG algorithm based on the well known Polak-Ribière-Polyak (PRP) formula for unconstrained optimization, where their method has the sufficient descent property without any line search technique. They proved the global convergence of the Armijo line search but this fails for the Wolfe line search technique. Inspired by their method, we will make a further study and give a modified three-term PRP CG algorithm. The presented method possesses the following features: (1) The sufficient descent property also holds without any line search technique; (2) the trust region property of the search direction is automatically satisfied; (3) the steplengh is bounded from below; (4) the global convergence will be established under the Wolfe line search. Numerical results show that the new algorithm is more effective than that of the normal method.

Keywords

conjugate gradient sufficient descent trust region

MSC

90C26

1 Introduction

We consider the optimization models defined by
$$ \min_{x\in\Re^{n}} f(x), $$
(1.1)
where the function \(f:\Re^{n}\rightarrow\Re\) is continuously differentiable. There exist many similar professional fields of science that can revert to the above optimization models (see, e.g., [221]). The CG method has the following iterative formula for (1.1):
$$ x_{k+1}=x_{k}+\alpha_{k} d_{k},\quad k=1, 2,\ldots, $$
(1.2)
where \(x_{k}\) is the kth iterate point, the steplength is \(\alpha_{k} > 0\), and the search direction \(d_{k}\) is designed by
$$\begin{aligned} d_{k+1}= \textstyle\begin{cases}-g_{k+1}+\beta_{k}d_{k}, & \mbox{if } k\geq1,\\ -g_{k+1},& \mbox{if }k=0, \end{cases}\displaystyle \end{aligned}$$
(1.3)
where \(g_{k}=\nabla f(x_{k})\) is the gradient and \(\beta_{k} \in\Re\) is a scalar. At present, there are many well-known CG formulas (see [2246]) and their applications (see, e.g., [4750]), where one of the most efficient formulas is the PRP [34, 51] defined by
$$ \beta_{k}^{\mathrm{PRP}}=\frac{g_{k+1}^{T}\delta_{k}}{ \Vert g_{k} \Vert ^{2}}, $$
(1.4)
where \(g_{k+1}=\nabla f(x_{k+1})\) is the gradient, \(\delta _{k}=g_{k+1}-g_{k}\), and \(\Vert . \Vert \) is the Euclidian norm. The PRP method is very efficient as regards numerical performance, but it fails as regards the global convergence for the general functions under Wolfe line search technique and this is a still open problem; many scholars want to solve it. It is worth noting that a recent work of Yuan et al. [52] proved the global convergence of PRP method under a modified Wolfe line search technique for general functions. Al-Baali [53], Gilbert and Nocedal [54], Toouati-Ahmed and Storey [55], and Hu and Storey [56] hinted that the sufficient descent property may be crucial for the global convergence of the conjugate gradient methods including the PRP method. Considering the above suggestions, Zhang, Zhou, and Li [1] firstly gave a three-term PRP formula
$$\begin{aligned} d_{k+1}= \textstyle\begin{cases} -g_{k+1}+\beta_{k}^{\mathrm{PRP}}d_{k}-\vartheta_{k}\delta_{k}, & \mbox{if } k\geq 1,\\ -g_{k+1},& \mbox{if } k=0, \end{cases}\displaystyle \end{aligned}$$
(1.5)
where \(\vartheta_{k}=\frac{g_{k+1}^{T}d_{k}}{ \Vert g_{k} \Vert ^{2}}\). It is not difficult to deduce that \(d_{k+1}^{T}g_{k+1}=- \Vert g_{k+1} \Vert ^{2}\) holds for all k, which implies that the sufficient descent property is satisfied. Zhang et al. proved that the three-term PRP method has global convergence under Armijo line search technique for general functions but this fails for the Wolfe line search. The reason may be the trust region feature of the search direction that cannot be satisfied for this method. In order to overcome this drawback, we will propose a modified three-term PRP formula that will have not only the sufficient descent property but also the trust region feature.

In the next section, a modified three-term PRP formula is given and the new algorithm is stated. The sufficient descent property, the trust region feature, and the global convergence of the new method are established in Section 3. Numerical results are reported in the last section.

2 The modified PRP formula and algorithm

Motivated by the above observation, the modified three-term PRP formula is
$$\begin{aligned} d_{k+1}= \textstyle\begin{cases} -g_{k+1}+\frac{g_{k+1}^{T}\delta_{k}d_{k}-d_{k}^{T}g_{k+1}\delta_{k}}{\gamma _{1} \Vert g_{k} \Vert ^{2}+\gamma_{2} \Vert d_{k} \Vert \delta_{k} \Vert +\gamma_{3} \Vert d_{k} \Vert g_{k} \Vert }, & \mbox{if } k\geq1,\\ -g_{k+1},& \mbox{if } k=0, \end{cases}\displaystyle \end{aligned}$$
(2.1)
where \(\gamma_{1}>0\), \(\gamma_{2}>0\), and \(\gamma_{3}>0\) are constants. It is easy to see that the difference between (1.5) and (2.1) is the denominator of the second and the third terms. This is a little change that will guarantee another good property for (2.1) and impel the global convergence for Wolfe conditions.

Algorithm 1

New three-term PRP CG algorithm (NTT-PRP-CG-A)

Step 0:: 

Initial given parameters: \(x_{1} \in \Re^{n}\), \(\gamma_{1}>0\), \(\gamma_{2}>0\), \(\gamma_{3}>0\), \(0<\delta<\sigma<1\), \(\varepsilon\in(0,1)\). Let \(d_{1}=-g_{1}=-\nabla f(x_{1})\) and \(k:=1\).

Step 1:: 

\(\Vert g_{k} \Vert \leq\varepsilon\), stop.

Step 2:: 
Get stepsize \(\alpha_{k}\) by the following Wolfe line search rules:
$$ f(x_{k}+\alpha_{k}d_{k}) \leq f(x_{k})+\delta\alpha_{k} g_{k}^{T}d_{k}, $$
(2.2)
and
$$ g(x_{k}+\alpha_{k}d_{k})^{T}d_{k} \geq\sigma g_{k}^{T}d_{k}. $$
(2.3)
Step 3:: 

Let \(x_{k+1}=x_{k}+\alpha_{k}d_{k}\). If the condition \(\Vert g_{k+1} \Vert \leq\varepsilon\) holds, stop the program.

Step 4:: 

Calculate the search direction \(d_{k+1}\) by (2.1).

Step 5:: 

Set \(k:=k+1\) and go to Step 2.

3 The sufficient descent property, the trust region feature, and the global convergence

It has been proved that, even for the function \(f(x)=\lambda \Vert x \Vert ^{2}\) (\(\lambda>0\) is a constant) and the strong Wolfe conditions, the PRP conjugate gradient method may not yield a descent direction for an unsuitable choice (see [24] for details). An interesting feature of the new three-term CG method is that the given search direction is sufficiently descent.

Lemma 3.1

The search direction \(d_{k}\) is defined by (2.1) and it satisfies
$$ d_{k+1}^{T}g_{k+1} = - \Vert g_{k+1} \Vert ^{2} $$
(3.1)
and
$$ \Vert d_{k+1} \Vert \leq\gamma \Vert g_{k+1} \Vert $$
(3.2)
for all \(k\geq0\), where \(\gamma>0\) is a constant.

Proof

For \(k=0\), it is easy to get \(g_{1}^{T}d_{1}=-g_{1}^{T}g_{1}=- \Vert g_{1} \Vert ^{2}\) and \(\Vert d_{1} \Vert = \Vert -g_{1} \Vert = \Vert g_{1} \Vert \), so (3.1) is true and (3.2) holds with \(\gamma= 1\).

If \(k\geq1\), by (2.1), we have
$$\begin{aligned} g_{k+1}^{T}d_{k+1} =& - \Vert g_{k+1} \Vert ^{2}+ g_{k+1}^{T}\biggl[ \frac {g_{k+1}^{T}\delta_{k}d_{k}-d_{k}^{T}g_{k+1}\delta_{k}}{\gamma_{1} \Vert g_{k} \Vert ^{2}+\gamma_{2} \Vert d_{k} \Vert \delta_{k} \Vert +\gamma_{3} \Vert d_{k} \Vert g_{k} \Vert }\biggr] \\ =& - \Vert g_{k+1} \Vert ^{2}+\frac{g_{k+1}^{T}\delta_{k} g_{k+1}^{T}d_{k}-d_{k}^{T}g_{k+1}g_{k+1}^{T} \delta_{k}}{\gamma_{1} \Vert g_{k} \Vert ^{2}+\gamma_{2} \Vert d_{k} \Vert \delta_{k} \Vert +\gamma_{3} \Vert d_{k} \Vert g_{k} \Vert } \\ =&- \Vert g_{k+1} \Vert ^{2}. \end{aligned}$$
(3.3)
Then (3.1) is satisfied. By (2.1) again, we obtain
$$ \begin{aligned}[b] \Vert d_{k+1} \Vert &= \biggl\Vert g_{k+1}+\frac{g_{k+1}^{T}\delta _{k}d_{k}-d_{k}^{T}g_{k+1}\delta_{k}}{\gamma_{1} \Vert g_{k}\Vert^{2}+\gamma_{2} \Vert d_{k}\Vert\delta_{k} \Vert+\gamma_{3}\Vert d_{k} \Vert g_{k}\Vert} \biggr\Vert \\ &\leq \Vert g_{k+1} \Vert+\frac{\Vert g_{k+1}^{T}\delta _{k}d_{k}-d_{k}^{T}g_{k+1} \delta_{k} \Vert}{\gamma_{1}\Vert g_{k} \Vert ^{2}+\gamma_{2}\Vert d_{k} \Vert\delta_{k}\Vert+\gamma_{3} \Vert d_{k}\Vert g_{k} \Vert} \\ &\leq\Vert g_{k+1} \Vert+\frac{\Vert\delta_{k} \Vert \Vert g_{k+1} \Vert\Vert d_{k} \Vert+\Vert d_{k} \Vert\Vert g_{k+1} \Vert\Vert \delta_{k} \Vert}{\gamma_{1}\Vert g_{k} \Vert^{2}+\gamma_{2}\Vert d_{k} \Vert \delta_{k}\Vert+\gamma_{3} \Vert d_{k}\Vert g_{k} \Vert} \\ &\leq\Vert g_{k+1} \Vert+\frac{2\Vert\delta_{k} \Vert \Vert g_{k+1} \Vert\Vert d_{k} \Vert}{\gamma_{2}\Vert d_{k} \Vert\delta_{k}\Vert } \\ &= (1+2/\gamma_{2})\Vert g_{k+1} \Vert, \end{aligned} $$
(3.4)
where the last inequality follows from \(\frac{1}{\gamma_{1} \Vert g_{k} \Vert ^{2}+\gamma_{2} \Vert d_{k} \Vert \delta_{k} \Vert +\gamma_{3} \Vert d_{k} \Vert g_{k} \Vert }\leq\frac {1}{\gamma_{2} \Vert d_{k} \Vert \delta_{k}\Vert}\). Thus (3.2) holds for all \(k\geq0\) with \(\gamma=\max\{1,1+2/\gamma_{2}\}\). The proof is complete. □

Remark

(1) Equation (3.1) is the sufficient descent property and the inequality (3.2) is the trust region feature. And these two relations are verifiable without any other conditions.

(2) Equations (3.1) and (2.2) imply that the sequence \(\{ f(x_{k})\}\) generated by Algorithm 1 is descent, namely \(f(x_{k}+\alpha _{k}d_{k})\leq f(x_{k})\) holds for all k.

To establish the global convergence of Algorithm 1, the normal conditions are needed.

Assumption A

  1. (i)

    The defined level set \(\Omega=\{x\in\Re^{n}\mid f(x)\leq f(x_{1})\}\) is bounded with given point \(x_{1}\).

     
  2. (ii)
    The function f has a lower bound and it is differentiable. The gradient g is Lipschitz continuous
    $$ \bigl\Vert g(x)-g(y) \bigr\Vert \leq L \Vert x-y \Vert , \quad \forall x,y\in\Re^{n}, $$
    (3.5)
    where \(L>0\) a constant.
     

Lemma 3.2

Suppose that Assumption A holds and NTT-PRP-CG-A generates the sequence \(\{x_{k},d_{k},\alpha_{k},g_{k}\}\). Then there exists a constant \(\beta >0\) such that
$$ \alpha_{k}\geq\beta,\quad\forall k\geq1. $$
(3.6)

Proof

Using (3.5) and (2.3) generate
$$\begin{aligned} \alpha_{k}L \geq& (g_{k+1}-g_{k})^{T}d_{k} \\ \geq& -(1-\sigma)g_{k}^{T}d_{k} \\ =& (1-\sigma) \Vert g_{k} \Vert ^{2}, \end{aligned}$$
where the last equality follows from (3.1). By (3.2), we get
$$\alpha_{k}\geq\frac{1-\sigma}{L}\frac{ \Vert g_{k} \Vert ^{2}}{ \Vert d_{k} \Vert ^{2}}\geq \frac{1-\sigma}{L\gamma}. $$
Setting \(\beta\in(0,\frac{1-\sigma}{L\gamma})\) completes the proof. □

Remark

The above lemma shows that the steplengh \(\alpha_{k}\) has a lower bound, which is helpful for the global convergence of Algorithm 1.

Theorem 3.1

Let the conditions of Lemma 3.2 hold and \(\{x_{k},d_{k},\alpha_{k},g_{k}\}\) be generated by NTT-PRP-CG-A. Thus we get
$$\lim_{k\rightarrow\infty} \Vert g_{k} \Vert =0. $$

Proof

By (2.2), (3.1), and (3.6), we have
$$\delta\beta \Vert g_{k} \Vert ^{2} \leq\delta \alpha_{k} \Vert g_{k} \Vert ^{2} \leq f(x_{k})-f(x_{k}+\alpha_{k}d_{k}). $$
Summing the above inequality from \(k=1\) to ∞, we have
$$\sum_{k=1}^{\infty}\delta\beta \Vert g_{k} \Vert ^{2} \leq f(x_{1})-f_{\infty}\leq\infty, $$
which means that
$$\Vert g_{k} \Vert \rightarrow0,\quad k\rightarrow\infty. $$
The proof is complete. □

4 Numerical results and discussion

This section will report the numerical experiment of the NTT-PRP-CG-A and the algorithm of Zhang et al. [1] (called Norm-PRP-A), where the Norm-PRP-A is the Step 4 of Algorithm 1 that is replaced by: Calculate the search direction \(d_{k+1}\) by (1.5). Since the method is based on the search direction (1.5), we only compare the numerical results between the new algorithm and the Norm-PRP-A. The codes of these two algorithms are written by Matlab and the problems are chosen from [57, 58] with given initial points. The tested problems are listed in Table 1. The parameters are \(\gamma_{1}=2\), \(\gamma_{2}=5\), \(\gamma_{3}=3\), \(\delta=0.01\), \(\sigma=0.86\). The program uses the Himmelblau rule: Set \(St_{1}=\frac{ \vert f(x_{k})-f(x_{k+1}) \vert }{ \vert f(x_{k}) \vert }\) if \(\vert f(x_{k}) \vert > \tau_{1}\), otherwise set \(St_{1}= \vert f(x_{k})-f(x_{k+1}) \vert \). The program stops if \(\Vert g(x) \Vert <\epsilon\) or \(St_{1} < \tau_{2}\) hold, where we choose \(\epsilon=10^{-6}\) and \(\tau_{1}=\tau _{2}=10^{-5}\). For the choice of the stepsize \(\alpha_{k}\) in (2.2) and (2.3), the largest cycle number is 10 and the current stepsize is accepted. The dimensions of the test problems accord to large-scale variables with 3,000, 12,000, and 30,000. The runtime environment is MATLAB R2010b and run on a PC with CPU Intel Pentium(R) Dual-Core CPU at 3.20 GHz, 2.00 GB of RAM, and the Windows 7 operating system.
Table 1

Test problems

No.

Problems

\(\boldsymbol{x_{0}}\)

1

Extended Freudenstein and Roth function

[0.5,−2,…,0.5,−2]

2

Extended trigonometric function

[0.2,0.2,…,0.2]

3

Extended Rosenbrock function

[−1.2,1,−1.2,1,…,−1.2,1]

4

Extended White and Holst function

[−1.2,1,−1.2,1,…,−1.2,1]

5

Extended Beale function

[1,0.8,…,1,0.8]

6

Extended penalty function

[1,2,3,…,n]

7

Perturbed quadratic function

[0.5,0.5,…,0.5]

8

Raydan 1 function

[1,1,…,1]

9

Raydan 2 function

[1,1,…,1]

10

Diagonal 1 function

[1/n,1/n,…,1/n]

11

Diagonal 2 function

[1/1,1/2,…,1/n]

12

Diagonal 3 function

[1,1,…,1]

13

Hager function

[1,1,…,1]

14

Generalized tridiagonal 1 function

[2,2,…,2]

15

Extended tridiagonal 1 function

[2,2,…,2]

16

Extended three exponential terms function

[0.1,0.1,…,0.1]

17

Generalized tridiagonal 2 function

[−1,−1,…,−1,−1]

18

Diagonal 4 function

[1,1,…,1,1]

19

Diagonal 5 function

[1.1,1.1,…,1.1]

20

Extended Himmelblau function

[1,1,…,1]

21

Generalized PSC1 function

[3,0.1,…,3,0.1]

22

Extended PSC1 function

[3,0.1,…,3,0.1]

23

Extended Powell function

[3,−1,0,1,…]

24

Extended block diagonal BD1 function

[0.1,0.1,…,0.1]

25

Extended Maratos function

[1.1,0.1,…,1.1,0.1]

26

Extended Cliff function

[0,−1,…,0,−1]

27

Quadratic diagonal perturbed function

[0.5,0.5,…,0.5]

28

Extended Wood function

[−3,−1,−3,−1,…,−3,−1]

29

Extended Hiebert function

[0,0,…,0]

30

Quadratic QF1 function

[1,1,…,1]

31

Extended quadratic penalty QP1 function

[1,1,…,1]

32

Extended quadratic penalty QP2 function

[1,1,…,1]

33

Quadratic QF2 function

[0.5,0.5,…,0.5]

34

Extended EP1 function

[1.5.,1.5.,…,1.5]

35

Extended tridiagonal-2 function

[1,1,…,1]

36

BDQRTIC function (CUTE)

[1,1,…,1]

37

TRIDIA function (CUTE)

[1,1,…,1]

38

ARWHEAD function (CUTE)

[1,1,…,1]

39

NONDIA (Shanno-78) function (CUTE)

[−1,−1,…,−1]

40

NONDQUAR function (CUTE)

[1,−1,1,−1,…,1,−1]

41

DQDRTIC function (CUTEr)

[3,3,3...,3]

42

EG2 function (CUTE)

[1,1,1...,1]

43

DIXMAANA function (CUTE)

[2,2,2,…,2]

44

DIXMAANB function (CUTE)

[2,2,2,…,2]

45

DIXMAANC function (CUTE)

[2,2,2,…,2]

46

DIXMAANE function (CUTE)

[2,2,2,…,2]

47

Partial perturbed quadratic function

[0.5,0.5,…,0.5]

48

Broyden tridiagonal function

[−1,−1,…,−1]

49

Almost perturbed quadratic function

[0.5,0.5,…,0.5]

50

Tridiagonal perturbed quadratic function

[0.5,0.5,…,0.5]

51

EDENSCH function (CUTE)

[0,0,…,0]

52

VARDIM function (CUTE)

[1 − 1/n,1 − 2/n,…,1 − n/n]

53

STAIRCASE S1 function

[1,1,…,1]

54

LIARWHD function (CUTEr)

[4,4,…,4]

55

DIAGONAL 6 function

[1,1,…,1]

56

DIXON3DQ function (CUTE)

[−1,−1,…,−1]

57

DIXMAANF function (CUTE)

[2,2,2,…,2]

58

DIXMAANG function (CUTE)

[2,2,2,…,2]

59

DIXMAANH function (CUTE)

[2,2,2,…,2]

60

DIXMAANI function (CUTE)

[2,2,2,…,2]

61

DIXMAANJ function (CUTE)

[2,2,2,…,2]

62

DIXMAANK function (CUTE)

[2,2,2,…,2]

63

DIXMAANL function (CUTE)

[2,2,2,…,2]

64

DIXMAAND function (CUTE)

[2,2,2,…,2]

65

ENGVAL1 function (CUTE)

[2,2,2,…,2]

66

FLETCHCR function (CUTE)

[0,0,…,0]

67

COSINE function (CUTE)

[1,1,…,1]

68

Extended DENSCHNB function (CUTE)

[1,1,…,1]

69

DENSCHNF function (CUTEr)

[2,0,2,0,…,2,0]

70

SINQUAD function (CUTE)

[0.1,0.1,…,0.1]

71

BIGGSB1 function (CUTE)

[0,0,…,0]

72

Partial perturbed quadratic PPQ2 function

[0.5,0.5,…,0.5]

73

Scaled quadratic SQ1 function

[1,2,…,n]

74

Scaled quadratic SQ2 function

[1,2,…,n]

Table 2 report the test numerical results of the NTT-PRP-CG-A and the Norm-PRP-A, and we notate:
Table 2

Numerical results

No.

Dimension

NTT-PRP-CG-A

Norm-PRP-A

Ni

Nfg

CPU time

Ni

Nfg

CPU time

1

3,000

15

43

0.468003

31

92

0.546004

12,000

15

43

0.842405

56

158

1.778411

30,000

15

43

1.482009

36

113

2.730018

2

3,000

57

131

0.374402

55

126

0.374402

12,000

63

144

1.138807

62

142

0.920406

30,000

66

152

3.08882

66

152

2.511616

3

3,000

54

186

0.124801

117

375

0.202801

12,000

67

233

0.234001

144

479

0.514803

30,000

73

238

0.530403

159

522

1.62241

4

3,000

59

198

0.296402

207

595

0.936006

12,000

34

139

0.733205

264

801

4.305628

30,000

74

256

4.118426

228

618

8.907657

5

3,000

23

68

0.093601

39

106

0.124801

12,000

23

69

0.265202

39

109

0.390003

30,000

21

64

0.826805

47

135

1.279208

6

3,000

80

185

0.124801

80

185

0.093601

12,000

103

232

0.405603

103

232

0.343202

30,000

102

235

1.216808

102

235

0.998406

7

3,000

1,000

2,002

1.045207

357

943

0.421203

12,000

1,000

2,002

3.16682

835

2,257

2.808018

30,000

1,000

2,002

9.781263

1,000

2,779

9.734462

8

3,000

21

47

0.0468

19

46

0.0312

12,000

20

44

0.093601

19

46

0.093601

30,000

20

44

0.296402

19

46

0.265202

9

3,000

12

26

0.0312

12

26

0.0312

12,000

12

26

0.0468

12

26

0.0624

30,000

12

26

0.202801

12

26

0.156001

10

3,000

2

13

0.0312

2

13

0.0312

12,000

2

13

0.124801

2

13

0.093601

30,000

2

13

0.312002

2

13

0.280802

11

3,000

81

194

0.171601

24

101

0.0624

12,000

91

247

0.764405

15

59

0.202801

30,000

11

35

0.436803

13

50

0.280802

12

3,000

17

36

0.0468

14

33

0.0624

12,000

19

40

0.171601

14

33

0.124801

30,000

19

40

0.499203

14

33

0.343202

13

3,000

23

86

0.093601

22

84

0.078

12,000

42

111

0.452403

42

111

0.468003

30,000

2

13

0.358802

2

13

0.327602

14

3,000

6

15

0.717605

6

15

0.733205

12,000

6

15

7.004445

5

13

5.709637

30,000

3

8

14.258491

3

8

13.587687

15

3,000

38

85

1.794011

66

176

3.04202

12,000

41

102

17.924515

60

169

28.09578

30,000

44

114

75.395283

68

194

120.245571

16

3,000

20

42

0.0624

20

42

0

12,000

24

50

0.171601

24

50

0.156001

30,000

24

50

0.483603

24

50

0.436803

17

3,000

24

55

0.156001

31

71

0.218401

12,000

33

73

0.764405

29

74

0.717605

30,000

48

103

3.042019

30

81

1.996813

18

3,000

3

10

0.0156

13

43

0.0312

12,000

3

10

0.0312

13

43

0.0156

30,000

3

10

0.0312

14

47

0.124801

19

3,000

3

9

0

3

9

0

12,000

3

9

0.0468

3

9

0.0312

30,000

3

9

0.124801

3

9

0.124801

20

3,000

33

82

0.0312

26

74

0.0312

12,000

11

61

0.0624

5

35

0.0312

30,000

5

35

0.093601

20

67

0.218401

21

3,000

25

59

0.093601

27

63

0.0624

12,000

27

63

0.249602

26

60

0.187201

30,000

25

58

0.530403

27

63

0.530403

22

3,000

6

31

0.0312

7

42

0

12,000

6

31

0.0624

5

21

0.0624

30,000

6

31

0.218401

5

21

0.124801

23

3,000

134

383

0.670804

334

986

1.52881

12,000

147

416

2.652017

452

1,309

7.73765

30,000

114

330

5.304034

291

854

12.776482

24

3,000

28

90

0.0624

50

126

0.109201

12,000

31

108

0.249602

60

146

0.405603

30,000

28

97

0.686404

67

160

1.170007

25

3,000

28

56

0.0312

28

56

0.0312

12,000

7

16

0.0156

231

774

0.748805

30,000

7

16

0.0312

213

774

2.028013

26

3,000

65

152

0.124801

65

152

0.124801

12,000

72

166

0.514803

72

166

0.468003

30,000

79

180

1.51321

79

180

1.341609

27

3,000

31

94

0.0624

104

327

0.156001

12,000

43

137

0.187201

202

655

0.639604

30,000

104

329

1.154407

384

1,231

4.024826

28

3,000

40

124

0.0468

31

76

0.0312

12,000

31

91

0.124801

38

95

0.124801

30,000

40

107

0.546003

32

78

0.265202

29

3,000

4

19

0.0312

100

287

0.124801

12,000

4

19

0.0156

84

240

0.312002

30,000

4

19

0.093601

93

264

0.842405

30

3,000

1,000

2,002

0.842405

446

1,205

0.436803

12,000

1,000

2,002

2.636417

754

2,010

2.074813

30,000

1,000

2,002

8.330453

1,000

2,721

8.065252

31

3,000

29

66

0.0468

29

66

0.0624

12,000

34

78

0.156001

34

78

0.156001

30,000

34

78

0.421203

34

78

0.452403

32

3,000

48

100

0.093601

48

100

0.093601

12,000

37

80

0.280802

37

80

0.234001

30,000

36

80

0.780005

36

80

0.670804

33

3,000

3

7

0

3

7

0

12,000

2

5

0

2

5

0.0312

30,000

2

5

0.0312

2

5

0

34

3,000

4

8

0.0312

4

8

0.0312

12,000

7

14

0.0624

7

14

0.0312

30,000

10

20

0.156001

10

20

0.124801

35

3,000

12

24

0.0312

12

24

0

12,000

21

42

0.093601

21

42

0.093601

30,000

4

10

0.093601

4

10

0.0312

36

3,000

14

48

1.138807

45

148

3.244821

12,000

8

28

6.910844

120

369

95.831414

30,000

17

55

55.427155

162

483

488.922734

37

3,000

776

1,559

0.733205

1,000

2,688

1.107607

12,000

1,000

2,006

3.322821

1,000

2,733

3.556823

30,000

1,000

2,011

9.828063

506

1,378

4.960832

38

3,000

9

30

0.0312

27

81

0.0312

12,000

10

32

0.0468

21

60

0.140401

30,000

11

34

0.140401

24

69

0.312002

39

3,000

26

52

0.0624

26

52

0

12,000

29

58

0.093601

29

58

0.093601

30,000

23

46

0.187201

23

46

0.171601

40

3,000

554

1,332

5.881238

1,000

2,856

11.013671

12,000

1,000

2,228

39.733455

1,000

2,892

43.352678

30,000

1,000

2,247

100.745446

1,000

2,866

108.186694

41

3,000

27

68

0.078

49

133

0.0312

12,000

28

69

0.093601

50

136

0.124801

30,000

37

91

0.390002

39

101

0.374402

42

3,000

6

24

0.0312

6

24

0

12,000

6

24

0.0624

6

24

0.0624

30,000

6

24

0.187201

6

24

0.156001

43

3,000

28

60

0.202801

28

60

0.218401

12,000

30

64

0.936006

30

64

0.858005

30,000

32

68

2.527216

31

66

2.230814

44

3,000

46

96

0.358802

46

96

0.296402

12,000

49

102

1.51321

49

102

1.404009

30,000

52

108

4.024826

52

108

3.728424

45

3,000

19

44

0.202801

19

44

0.124801

12,000

20

46

0.608404

20

46

0.577204

30,000

20

46

1.54441

20

46

1.48201

46

3,000

117

244

0.920406

108

296

0.967206

12,000

165

340

5.116833

120

326

4.009226

30,000

195

400

15.678101

126

341

10.576868

47

3,000

27

66

8.299253

44

102

12.963683

12,000

31

87

93.741001

49

141

150.150963

30,000

69

182

1,163.84546

85

256

1,490.683156

48

3,000

32

74

1.762811

27

63

1.310408

12,000

50

103

30.154993

29

74

19.546925

30,000

42

100

112.726323

37

87

94.209004

49

3,000

1,000

2,002

0.858005

575

1,593

0.577204

12,000

1,000

2,002

2.792418

885

2,377

2.527216

30,000

1,000

2,002

9.484861

1,000

2,738

8.143252

50

3,000

1,000

2,002

57.236767

370

998

23.727752

12,000

1,000

2,002

617.73276

920

2,495

676.233135

30,000

1,000

2,002

2,467.96702

1,000

2,720

2,856.471911

51

3,000

23

48

0.124801

23

48

0.140401

12,000

23

48

0.811205

23

48

0.452403

30,000

23

48

1.154407

23

48

1.216808

52

3,000

121

276

0.436803

121

276

0.374402

12,000

138

316

2.090413

138

316

1.684811

30,000

150

344

4.66443

150

344

4.61763

53

3,000

1,000

2,009

0.998406

1,000

2,706

1.170008

12,000

1,000

2,009

3.759624

1,000

2,661

3.369622

30,000

1,000

2,009

8.502054

1,000

2,781

9.594061

54

3,000

32

87

0.0624

203

577

0.343202

12,000

13

41

0.109201

201

607

1.029607

30,000

42

99

0.483603

362

1,112

4.836031

55

3,000

21

44

0.546004

21

44

0.530403

12,000

23

48

7.488048

23

48

7.441248

30,000

24

50

30.654197

24

50

29.983392

56

3,000

430

886

0.358802

507

1,397

0.608404

12,000

430

886

1.450809

613

1,667

2.043613

30,000

430

886

3.541223

491

1,337

4.492829

57

3,000

145

296

1.154407

55

132

0.468003

12,000

207

420

7.75325

69

179

2.246414

30,000

265

536

19.500125

77

196

6.27124

58

3,000

107

223

0.873606

81

202

0.670804

12,000

124

257

3.931225

91

243

3.07322

30,000

142

293

10.514467

98

261

8.205653

59

3,000

77

166

0.639604

52

137

0.405603

12,000

107

226

4.508429

60

152

1.934412

30,000

94

203

7.082445

72

181

5.803237

60

3,000

488

983

3.978026

111

303

0.967206

12,000

175

360

5.522435

106

293

3.650423

30,000

194

398

14.476893

140

377

11.856076

61

3,000

145

296

1.185608

56

142

0.468003

12,000

206

418

6.692443

70

179

2.277615

30,000

264

534

19.390924

92

247

7.75325

62

3,000

153

314

1.232408

63

163

0.717605

12,000

239

486

7.332047

86

214

2.761218

30,000

313

634

23.166148

96

261

8.127652

63

3,000

209

430

1.934412

138

378

1.388409

12,000

1,000

2,009

37.799042

164

448

6.489642

30,000

1,000

2,009

87.220159

191

521

18.532919

64

3,000

29

64

0.265202

29

64

0.218401

12,000

31

68

1.045207

31

68

0.936006

30,000

32

70

2.340015

32

70

2.324415

65

3,000

22

51

1.903212

19

45

1.59121

12,000

17

38

14.586094

17

38

14.258491

30,000

17

38

61.167992

17

38

59.420781

66

3,000

1,000

2,003

57.985572

733

2,293

50.684725

12,000

1,000

2,003

618.637566

214

671

171.757101

30,000

4

11

10.374067

58

157

163.879051

67

3,000

6

37

0.0312

9

59

0.0312

12,000

10

63

0.499203

48

231

0.577204

30,000

5

27

0.124801

10

54

0.296402

68

3,000

35

72

0.0312

35

72

0.0312

12,000

38

78

0.124801

38

78

0.109201

30,000

39

80

0.343202

39

80

0.374402

69

3,000

27

58

0.0312

30

64

0.0624

12,000

28

60

0.140401

32

68

0.187201

30,000

29

62

0.421203

33

70

0.468003

70

3,000

25

82

1.950013

129

386

8.876457

12,000

52

184

46.8471

143

479

119.621567

30,000

13

62

52.790738

193

598

597.967433

71

3,000

1,000

2,004

0.889206

449

1,247

0.468003

12,000

1,000

2,004

4.196427

661

1,779

2.106014

30,000

1,000

2,004

7.238446

606

1,645

5.506835

72

3,000

706

2,011

228.837867

1,000

2,845

323.405673

12,000

569

1,589

1,742.46877

785

2,234

2,412.040662

30,000

229

654

3,931.381201

1,000

2,813

17,084.27791

73

3,000

1,000

2,002

0.936006

490

1,307

0.421203

12,000

1,000

2,002

3.291621

900

2,460

2.605217

30,000

1,000

2,002

7.566048

1,000

2,735

7.940451

74

3,000

1,000

2,002

0.873606

398

1,061

0.374402

12,000

1,000

2,002

4.399228

795

2,120

2.262015

30,000

1,000

2,002

7.519248

1,000

2,682

7.86245

No. the test problems number. Dimension: the variables number.

Ni: the iteration number. Nfg: the function and the gradient value number. CPU time: the CPU time of operating system in seconds.

A new tool was given by Dolan and Moré [59] to analyze the performance of the algorithms. Figures 1-3 show that the efficiency of the NTT-PRP-CG-A and the Norm-PRP-A relate to Ni, Nfg, and CPU time, respectively. It is easy to see that these two algorithms are effective for those problems and the given three-term PRP conjugate gradient method is more effective than that of the normal three-term PRP conjugate gradient method. Moreover, the NTT-PRP-CG-A has good robustness. Overall, the presented algorithm has some potential property both in theory and numerical experiment, which is noticeable.
Figure 1

Performance profiles of the algorithms for the test problems (Ni).

Figure 2

Performance profiles of the algorithms for the test problems (Nfg).

Figure 3

Performance profiles of the algorithms for the test problems (CPU time).

5 Conclusions

In this paper, based on the PRP formula for unconstrained optimization, a modified three-term PRP CG algorithm was presented. The proposed method possesses sufficient descent property also holds without any line search technique, and we have automatically the trust region property of the search direction. Under the Wolfe line search, the global convergence was proven. Numerical results showed that the new algorithm is more effective compared with the normal method.

Declarations

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Basic Teaching, Yango College

References

  1. Zhang, L, Zhou, W, Li, D: A descent modified Polak-Ribière-Polyak conjugate method and its global convergence. IMA J. Numer. Anal. 26, 629-649 (2006) MathSciNetView ArticleMATHGoogle Scholar
  2. Fu, Z, Wu, X, Guan, C, Sun, X, Ren, K: Towards efficient multi-keyword fuzzy search over encrypted outsourced data with accuracy improvement. IEEE Trans. Inf. Forensics Secur. (2016). doi:https://doi.org/10.1109/TIFS.2016.2596138 Google Scholar
  3. Gu, B, Sheng, VS, Tay, KY, Romano, W, Li, S: Incremental support vector learning for ordinal regression. IEEE Trans. Neural Netw. Learn. Syst. 26, 1403-1416 (2015) MathSciNetView ArticleGoogle Scholar
  4. Gu, B, Sun, X, Sheng, VS: Structural minimax probability machine, IEEE Trans. Neural Netw. Learn. Syst. (2016). doi:https://doi.org/10.1109/TNNLS.2016.2544779 Google Scholar
  5. Li, J, Li, X, Yang, B, Sun, X: Segmentation-based image copy-move forgery detection scheme. IEEE Trans. Inf. Forensics Secur. 10, 507-518 (2015) View ArticleGoogle Scholar
  6. Pan, Z, Lei, J, Zhang, Y, Sun, X, Kwong, S: Fast motion estimation based on content property for low-complexity H.265/HEVC encoder. IEEE Trans. Broadcast. (2016). doi:https://doi.org/10.1109/TBC.2016.2580920 Google Scholar
  7. Pan, Z, Zhang, Y, Kwong, S: Efficient motion and disparity estimation optimization for low complexity multiview video coding. IEEE Trans. Broadcast. 61, 166-176 (2015) View ArticleGoogle Scholar
  8. Xia, Z, Wang, X, Sun, X, Wang, Q: A secure and dynamic multi-keyword ranked search scheme over encrypted cloud data. IEEE Trans. Parallel Distrib. Syst. 27, 340-352 (2015) View ArticleGoogle Scholar
  9. Xia, Z, Wang, X, Sun, X, Liu, Q, Xiong, N: Steg analysis of LSB matching using differences between nonadjacent pixels. Multimed. Tools Appl. 75, 1947-1962 (2016) View ArticleGoogle Scholar
  10. Xia, Z, Wang, X, Zhang, L, Qin, Z, Sun, X, Ren, K: A privacy-preserving and copy-deterrence content-based image retrieval scheme in cloud computing. IEEE Trans. Inf. Forensics Secur. (2016). doi:https://doi.org/10.1109/TIFS.2016.2590944 Google Scholar
  11. Yuan, GL: A new method with descent property for symmetric nonlinear equations. Numer. Funct. Anal. Optim. 31, 974-987 (2010) MathSciNetView ArticleMATHGoogle Scholar
  12. Yuan, GL, Lu, S, Wei, ZX: A new trust-region method with line search for solving symmetric nonlinear equations. Int. J. Comput. Math. 88, 2109-2123 (2011) MathSciNetView ArticleMATHGoogle Scholar
  13. Yuan, C, Sun, X, Lv, R: Fingerprint liveness detection based on multi-scale LPQ and PCA. China Commun. 13, 60-65 (2016) View ArticleGoogle Scholar
  14. Yuan, GL, Lu, XW, Wei, ZX: BFGS trust-region method for symmetric nonlinear equations. J. Comput. Appl. Math. 230, 44-58 (2009) MathSciNetView ArticleMATHGoogle Scholar
  15. Yuan, G, Wei, Z: A trust region algorithm with conjugate gradient technique for optimization problems. Numer. Funct. Anal. Optim. 32, 212-232 (2011) MathSciNetView ArticleMATHGoogle Scholar
  16. Yuan, GL, Wei, ZX, Lu, S: Limited memory BFGS method with backtracking for symmetric nonlinear equations. Math. Comput. Model. 54, 367-377 (2011) MathSciNetView ArticleMATHGoogle Scholar
  17. Yuan, GL, Wei, ZX, Lu, XW: A BFGS trust-region method for nonlinear equations. Computing 92, 317-333 (2011) MathSciNetView ArticleMATHGoogle Scholar
  18. Yuan, GL, Wei, ZX, Wang, ZX: Gradient trust region algorithm with limited memory BFGS update for nonsmooth convex minimization. Comput. Optim. Appl. 54, 45-64 (2013) MathSciNetView ArticleMATHGoogle Scholar
  19. Yuan, GL, Wei, ZX, Wu, YL: Modified limited memory BFGS method with nonmonotone line search for unconstrained optimization. J. Korean Math. Soc. 47, 767-788 (2010) MathSciNetView ArticleMATHGoogle Scholar
  20. Yuan, GL, Yao, SW: A BFGS algorithm for solving symmetric nonlinear equations. Optimization 62, 45-64 (2013) MathSciNetView ArticleMATHGoogle Scholar
  21. Zhou, Z, Wang, Y, Wu, QMJ, Yang, C-N, Sun, X: Effective and efficient global context verification for image copy detection. IEEE Trans. Inf. Forensics Secur. (2016). doi:https://doi.org/10.1109/TIFS.2016.2601065 Google Scholar
  22. Andrei, N: A hybrid conjugate gradient algorithm for unconstrained optimization as a convex combination of Hestenes-Stiefel and Dai-Yuan. Stud. Inform. Control 17, 55-70 (2008) Google Scholar
  23. Dai, Y: A nonmonotone conjugate gradient algorithm for unconstrained optimization. J. Syst. Sci. Complex. 15, 139-145 (2002) MathSciNetMATHGoogle Scholar
  24. Dai, Y, Yuan, Y: A nonlinear conjugate gradient with a strong global convergence properties. SIAM J. Optim. 10, 177-182 (2000) View ArticleMATHGoogle Scholar
  25. Dai, Y, Yuan, Y: Nonlinear Conjugate Gradient Methods. Shanghai Sci. Technol., Shanghai (1998) Google Scholar
  26. Dai, Y, Yuan, Y: An efficient hybrid conjugate gradient method for unconstrained optimization. Ann. Oper. Res. 103, 33-47 (2001) MathSciNetView ArticleMATHGoogle Scholar
  27. Fletcher, R: Practical Method of Optimization, Vol I: Unconstrained Optimization, 2nd edn. Wiley, New York (1997) Google Scholar
  28. Fletcher, R, Reeves, C: Function minimization by conjugate gradients. Comput. J. 7, 149-154 (1964) MathSciNetView ArticleMATHGoogle Scholar
  29. Grippo, L, Lucidi, S: A globally convergent version of the Polak-Ribière gradient method. Math. Program. 78, 375-391 (1997) MATHGoogle Scholar
  30. Hager, WW, Zhang, H: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16, 170-192 (2005) MathSciNetView ArticleMATHGoogle Scholar
  31. Hager, WW, Zhang, H: Algorithm 851: CG-DESCENT, a conjugate gradient method with guaranteed descent. ACM Trans. Math. Softw. 32, 113-137 (2006) MathSciNetView ArticleMATHGoogle Scholar
  32. Hager, WW, Zhang, H: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2, 35-58 (2006) MathSciNetMATHGoogle Scholar
  33. Hestenes, MR, Stiefel, E: Method of conjugate gradient for solving linear equations. J. Res. Natl. Bur. Stand. 49, 409-436 (1952) View ArticleMATHGoogle Scholar
  34. Polak, E, Ribière, G: Note sur la convergence de directions conjugees. Rev. Fr. Autom. Inform. Rech. Opér. 3, 35-43 (1969) MATHGoogle Scholar
  35. Powell, MJD: Nonconvex minimization calculations and the conjugate gradient method. In: Numerical Analysis, Lecture Notes in Mathematics, vol. 1066, pp. 122-141. Spinger, Berlin (1984) View ArticleGoogle Scholar
  36. Powell, MJD: Convergence properties of algorithm for nonlinear optimization. SIAM Rev. 28, 487-500 (1986) MathSciNetView ArticleMATHGoogle Scholar
  37. Wei, Z, Li, G, Qi, L: New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems. Appl. Math. Comput. 179, 407-430 (2006) MathSciNetMATHGoogle Scholar
  38. Wei, Z, Li, G, Qi, L: Global convergence of the PRP conjugate gradient methods with inexact line search for nonconvex unconstrained optimization problems. Math. Comput. 77, 2173-2193 (2008) View ArticleMATHGoogle Scholar
  39. Wei, Z, Yao, S, Liu, L: The convergence properties of some new conjugate gradient methods. Appl. Math. Comput. 183, 1341-1350 (2006) MathSciNetMATHGoogle Scholar
  40. Yuan, GL: Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems. Optim. Lett. 3, 11-21 (2009) MathSciNetView ArticleMATHGoogle Scholar
  41. Yuan, GL, Duan, XB, Liu, WJ, Wang, XL, Cui, ZR, Sheng, Z: Two new PRP conjugate gradient algorithms for minimization optimization models. PLoS ONE 10(10), e0140071 (2015) View ArticleGoogle Scholar
  42. Yuan, GL, Lu, XW: A modified PRP conjugate gradient method. Ann. Oper. Res. 166, 73-90 (2009) MathSciNetView ArticleMATHGoogle Scholar
  43. Yuan, GL, Lu, XW, Wei, ZX: A conjugate gradient method with descent direction for unconstrained optimization. J. Comput. Appl. Math. 233, 519-530 (2009) MathSciNetView ArticleMATHGoogle Scholar
  44. Yuan, GL, Wei, ZX: New line search methods for unconstrained optimization. J. Korean Stat. Soc. 38, 29-39 (2009) MathSciNetView ArticleMATHGoogle Scholar
  45. Yuan, GL, Wei, ZX, Zhao, QM: A modified Polak-Ribière-Polyak conjugate gradient algorithm for large-scale optimization problems. IIE Trans. 46, 397-413 (2014) View ArticleGoogle Scholar
  46. Yuan, GL, Zhang, MJ: A modified Hestenes-Stiefel conjugate gradient algorithm for large-scale optimization. Numer. Funct. Anal. Optim. 34, 914-937 (2013) MathSciNetView ArticleMATHGoogle Scholar
  47. Yuan, GL, Meng, ZH, Li, Y: A modified Hestenes and Stiefel conjugate gradient algorithm for large-scale nonsmooth minimizations and nonlinear equations. J. Optim. Theory Appl. 168, 129-152 (2016) MathSciNetView ArticleMATHGoogle Scholar
  48. Yuan, GL, Wei, ZX: The Barzilai and Borwein gradient method with nonmonotone line search for nonsmooth convex optimization problems. Math. Model. Anal. 17, 203-216 (2012) MathSciNetView ArticleMATHGoogle Scholar
  49. Yuan, GL, Wei, ZX, Li, GY: A modified Polak-Ribière-Polyak conjugate gradient algorithm for nonsmooth convex programs. J. Comput. Appl. Math. 255, 86-96 (2014) MathSciNetView ArticleMATHGoogle Scholar
  50. Yuan, GL, Zhang, MJ: A three-terms Polak-Ribière-Polyak conjugate gradient algorithm for large-scale nonlinear equations. J. Comput. Appl. Math. 286, 186-195 (2015) MathSciNetView ArticleMATHGoogle Scholar
  51. Polak, E: The conjugate gradient method in extreme problems. Comput. Math. Math. Phys. 9, 94-112 (1969) View ArticleGoogle Scholar
  52. Yuan, GL, Wei, ZX, Lu, XW: Global convergence of BFGS and PRP methods under a modified weak Wolfe-Powell line search. Appl. Math. Model. (2017). doi:https://doi.org/10.1016/j.apm.2017.02.008 Google Scholar
  53. Al-Baali, A: Descent property and global convergence of the Flecher-Reeves method with inexact line search. IMA J. Numer. Anal. 5, 121-124 (1985) MathSciNetView ArticleMATHGoogle Scholar
  54. Gilbert, JC, Nocedal, J: Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim. 2, 21-42 (1992) MathSciNetView ArticleMATHGoogle Scholar
  55. Ahmed, T, Storey, D: Efficient hybrid conjugate gradient techniques. J. Optim. Theory Appl. 64, 379-394 (1990) MathSciNetView ArticleMATHGoogle Scholar
  56. Hu, YF, Storey, C: Global convergence result for conjugate method. J. Optim. Theory Appl. 71, 399-405 (1991) MathSciNetView ArticleMATHGoogle Scholar
  57. Bongartz, I, Conn, AR, Gould, NI, Toint, PL: CUTE: constrained and unconstrained testing environment. ACM Trans. Math. Softw. 21, 123-160 (1995) View ArticleMATHGoogle Scholar
  58. Gould, NI, Orban, D, Toint, PL: CUTEr and SifDec: a constrained and unconstrained testing environment, revised. ACM Trans. Math. Softw. 29, 373-394 (2003) View ArticleMATHGoogle Scholar
  59. Dolan, ED, Moré, JJ: Benchmarking optimization software with performance profiles. Math. Program. 91, 201-213 (2002) MathSciNetView ArticleMATHGoogle Scholar

Copyright

© The Author(s) 2017