A modified threeterm PRP conjugate gradient algorithm for optimization models
 Yanlin Wu^{1}Email author
DOI: 10.1186/s1366001713734
© The Author(s) 2017
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 largescale problems, because of its low memory requirement and simplicity. Zhang et al. (IMA J. Numer. Anal. 26:629649, 2006) firstly propose a threeterm CG algorithm based on the well known PolakRibièrePolyak (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 threeterm 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 regionMSC
90C261 Introduction
In the next section, a modified threeterm 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
Algorithm 1
New threeterm PRP CG algorithm (NTTPRPCGA)
 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:and$$ f(x_{k}+\alpha_{k}d_{k}) \leq f(x_{k})+\delta\alpha_{k} g_{k}^{T}d_{k}, $$(2.2)$$ g(x_{k}+\alpha_{k}d_{k})^{T}d_{k} \geq\sigma g_{k}^{T}d_{k}. $$(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 threeterm CG method is that the given search direction is sufficiently descent.
Lemma 3.1
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 \(\gamma= 1\).
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
 (i)
The defined level set \(\Omega=\{x\in\Re^{n}\mid f(x)\leq f(x_{1})\}\) is bounded with given point \(x_{1}\).
 (ii)The function f has a lower bound and it is differentiable. The gradient g is Lipschitz continuouswhere \(L>0\) a constant.$$ \bigl\Vert g(x)g(y) \bigr\Vert \leq L \Vert xy \Vert , \quad \forall x,y\in\Re^{n}, $$(3.5)
Lemma 3.2
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
Proof
4 Numerical results and discussion
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 tridiagonal2 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 (Shanno78) 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] 
Numerical results
No.  Dimension  NTTPRPCGA  NormPRPA  

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.
5 Conclusions
In this paper, based on the PRP formula for unconstrained optimization, a modified threeterm 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
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