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A nonmonotone hybrid conjugate gradient method for unconstrained optimization
 Wenyu Li^{1} and
 Yueting Yang^{1}Email author
https://doi.org/10.1186/s1366001506441
© Li and Yang; licensee Springer. 2015
 Received: 16 January 2015
 Accepted: 27 March 2015
 Published: 8 April 2015
Abstract
A nonmonotone hybrid conjugate gradient method is proposed, in which the technique of the nonmonotone Wolfe line search is used. Under mild assumptions, we prove the global convergence and linear convergence rate of the method. Numerical experiments are reported.
Keywords
 unconstrained optimization
 nonmonotone hybrid conjugate gradient algorithm
 global convergence
 linear convergence rate
1 Introduction
The aim of this paper is to propose a nonmonotone hybrid conjugate gradient method which combines the nonmonotone line search technique with the LY method. It is based on the idea that the larger values of the stepsize \(\alpha_{k}\) may be accepted by the nonmonotone algorithmic framework and improve the behavior of the LY method.
The paper is organized as follows. A new nonmonotone hybrid conjugate gradient algorithm is presented and the global convergence of the algorithm is proved in Section 2. The line convergence rate of the algorithm is shown in Section 3. In Section 4, numerical results are reported.
2 Nonmonotone hybrid conjugate gradient algorithm and global convergence
Now we present a nonmonotone hybrid conjugate gradient algorithm.
Algorithm 1

Step 1. Given \(x_{0}\in R^{n}\), \(\epsilon>0\), \(d_{0}=g_{0}\), \(C_{0}=f_{0}\), \(Q_{0}, \zeta_{0}, k:=0\).

Step 2. If \(\g_{k}\<\epsilon\), then stop. Otherwise, compute \(\alpha_{k}\) by (9) and (7), set \(x_{k+1}=x_{k}+\alpha_{k}d_{k}\).

Step 3. Compute \(\beta_{k+1}\) by (2), set \(d_{k+1}=g_{k+1}+\beta_{k+1}d_{k}\), \(k:=k+1\), and go to Step 2.
Assumption 1
 (i)
The level set \({\Omega_{0}}=\{x\in R^{n}:{f{(x)}\leq f{(x_{0})}}\}\) is bounded, where \(x_{0}\) is the initial point.
 (ii)The gradient function \(g(x)=\nabla f(x)\) of the objective function f is Lipschitz continuous in a neighborhood \(\mathcal{N}\) of level set \({\Omega_{0}}\), i.e. there exists a constant \(L\geq0\) such thatfor any \({x,\bar{x}}\in\mathcal{N}\).$$\bigl\ g{(x)}g{(\bar{x})}\bigr\ \leq L\x\bar{x}\, $$
Lemma 2.1
Let the sequence \(\{x_{k}\}\) be generated by Algorithm 1. Then \(d_{k}^{T}g_{k}<0\) holds for all \(k\geq1\).
Proof
From Lemma 2 and Lemma 3 in [5], the conclusion holds. □
Lemma 2.2
Proof
Lemma 2.3
Proof
See Lemma 1.1 in [10]. □
Lemma 2.4
Proof
Theorem 2.1
Proof
3 Linear convergence rate of algorithm
Lemma 3.1
Suppose that the sequence \(\{x_{k}\}\) is generated by Algorithm 1 with the strong Wolfe line search (9) and (18), \(0<\sigma <\frac{\lambda}{1+\mu}\). Then there exists some constant \(c>0\) such that the direction condition (20) holds.
Proof
From (23) and (25), we obtain (20), where \(c=\min\{1\sigma\mu, 1\frac{\mu\sigma}{\lambda\sigma}\}>0\). The proof is completed. □
Lemma 3.2
Proof
Theorem 3.1
Proof
4 Numerical experiments
Test problems
No.  Problem name 

1  Helical valley function 
2  BIGGS6 
3  Gaussian function 
4  POWELLBS 
5  BOX3 
6  BEALE 
7  WOODS 
8  FREUROTH 
9  Osborne 1 function 
10  Osborne 2 function 
11  Powell singular function 
12  Meyer function 
13  Bard function 
14  PENALTY2 
15  VARDIM 
16  PENALTY1 
17  EXTROSNB 
18  Extended Powell singular function 
19  CHEBYQAD 
20  BROYDN3D 
21  Separable cubic function 
22  Nearly separable function 
23  Allgower function 
24  Schittkowski function 302 
25  Discrete integral equation function 
26  BDQRTIC 
27  ARGLINB 
28  ARWHEAD 
29  NONDIA 
30  NONDQUAR 
31  DQDRTIC 
32  EG2 
33  CURLY20 
34  LIARWHD 
35  ENGVAL1 
36  CRAGGLVY 
37  the Edensch function 
38  the Explin 1 function 
39  the Argling function 
40  NONSCOMP 
Numerical comparisons of LY, NMLY1, and NMLY2
No.  Dim  LY  NMLY2  NMLY1  

it  nf  ng  t  it  nf  ng  t  it  nf  ng  t  
1  3  36  69  48  0.0156  59  137  92  0.0156  40  84  60  0.0156 
2  6  147  243  200  0.0468  413  853  697  0.1404  146  270  213  0.0468 
3  3  2  5  4  0  4  10  8  0  3  9  8  0 
4  2  9  26  13  0  11  48  28  0  13  38  17  0.0312 
5  3  15  28  19  0  29  75  68  0.0312  14  35  28  0 
6  2  37  61  46  0.0156  28  64  45  0  20  60  30  0.0156 
7  4  74  130  100  0.0156  313  623  458  0.0936  62  119  90  0.0156 
8  2  44  82  61  0.0156  37  98  70  0  5  25  9  0.0156 
9  5  128  233  184  0.0936  181  383  278  0.1404  104  199  153  0.078 
10  11  3  42  3  0.0156  4  29  4  0  3  13  3  0.0312 
11  4  6  49  7  0  169  348  254  0.0468  59  121  77  0.0312 
12  3  9  45  14  0  170  456  328  0.1248  13  80  34  0.0156 
13  3  25  49  33  0.0156  64  135  101  0.0468  39  75  52  0.0312 
14  500  14  70  28  0.0156  10  82  40  0.0156  3  46  10  0.0156 
15  500  13  68  14  0  13  118  48  0.0312  17  187  122  0.0156 
1,000  16  70  16  0  13  87  35  0.0156  19  301  211  0.0312  
2,000  14  58  14  0.0156  15  116  50  0.0156  26  362  237  0.0624  
16  500  11  35  22  0.0156  40  147  101  0.0156  34  103  87  0.0156 
1,000  16  40  23  0  49  141  99  0.0312  21  82  62  0.0156  
2,000  20  51  28  0.0312  34  122  73  0.0312  57  156  108  0.0624  
1,000  22  75  30  0.078  28  96  67  0.1248  31  134  70  0.156  
17  500  42  87  58  0.0156  133  294  213  0.078  3  18  3  0 
1,000  64  120  87  0.0312  187  406  299  0.0936  3  19  3  0  
2,000  53  105  72  0.0312  134  304  221  0.0936  3  19  3  0.0156  
1,000  56  113  79  0.2028  184  393  292  0.624  3  19  3  0.0312  
18  500  220  364  295  0.1092  184  353  253  0.1092  110  197  160  0.0624 
1,000  180  288  238  0.1404  163  322  235  0.1404  104  191  150  0.078  
2,000  181  308  248  0.2184  181  361  262  0.2496  112  205  164  0.156  
1,000  217  361  293  1.2792  320  625  458  2.1372  133  244  197  0.8892  
19  500  68  105  86  5.694  103  210  146  10.9201  70  112  95  6.1308 
1,000  99  143  125  32.7602  88  185  133  38.891  113  201  144  40.0923  
2,000  72  110  95  90.5742  190  387  287  306.2456  78  141  104  110.3239  
1,000  121  197  142  983.8203  35  156  80  708.6189  24  67  45  322.8129  
20  500  51  74  60  0.0468  55  114  73  0.0156  53  83  68  0.0312 
1,000  54  78  62  0.0312  55  117  73  0.0312  93  143  115  0.0468  
2,000  44  74  58  0.0312  58  130  87  0.0624  33  61  39  0.0156  
1,000  43  69  57  0.1872  52  115  71  0.1872  42  75  59  0.1248  
21  500  4  9  8  0.0468  10  20  19  0.1092  8  18  17  0.1092 
1,000  5  11  10  0.234  10  20  19  0.468  9  20  19  0.468  
2,000  5  11  10  0.9204  10  20  19  1.8408  9  20  19  1.8252  
1,000  5  11  10  6.63  11  23  22  15.3973  10  21  20  13.8997  
22  500  50  91  68  1.1076  86  207  133  2.2308  36  92  65  1.0764 
1,000  33  96  52  3.3852  52  146  100  6.552  43  104  71  4.602  
2,000  27  113  48  12.1681  43  147  91  23.6186  52  167  83  22.9477  
1,000  102  222  142  246.5128  76  196  140  244.2664  42  123  71  124.442  
23  500  4  32  8  0.1092  13  69  30  0.4368  3  36  6  0.078 
1,000  6  69  9  0.4836  17  230  61  3.4944  3  57  5  0.234  
2,000  5  61  8  1.794  16  165  41  10.3585  3  37  3  0.5304  
1,000  5  62  11  16.1617  12  125  20  30.6074  3  56  5  6.1932  
24  500  59  137  88  0.0156  62  232  133  0.0312  34  192  121  0.0156 
1,000  26  91  34  0.0156  34  174  98  0.0312  56  292  204  0.0624  
2,000  17  68  26  0.0156  16  112  35  0.0312  22  272  172  0.0624  
1,000  37  89  42  0.1092  34  164  59  0.156  25  266  146  0.2496  
25  500  7  67  43  12.8545  13  85  53  15.9589  13  80  26  10.4521 
1,000  7  59  28  26.7698  8  78  51  41.9019  47  229  130  115.8931  
2,000  22  141  90  275.5914  7  55  28  96.081  7  68  28  107.2507  
1,000  7  63  36  716.4346  16  175  117  2148.9606  11  115  63  1265.6829  
26  500  54  105  77  1.6224  55  127  88  1.7628  46  97  68  1.3728 
1,000  53  101  75  5.4132  92  195  140  10.4053  46  94  69  5.0856  
2,000  42  89  62  15.1945  71  162  113  27.5654  40  92  65  16.0213  
1,000  29  68  43  44.4603  58  150  105  110.0275  44  97  68  69.7168  
27  500  3  52  4  0.2184  11  178  91  1.9968  9  152  12  0.6708 
1,000  13  193  99  8.7049  5  68  13  1.5756  7  106  27  2.9484  
2,000  5  101  46  16.3645  7  141  40  16.5205  7  141  48  18.7045  
1,000  9  112  32  99.279  8  140  33  112.5859  13  224  48  172.6151  
28  500  8  23  10  0  22  60  38  0.0468  19  45  27  0.0156 
1,000  9  25  13  0.0156  7  45  31  0.0156  11  59  43  0.0312  
2,000  16  60  26  0.0312  13  78  42  0.0624  16  72  36  0.0468  
1,000  5  22  7  0.0312  10  53  30  0.2496  11  60  40  0.2808  
29  500  28  65  41  0.0468  13  69  52  0.0312  13  50  27  0.0312 
1,000  4  17  5  0  22  86  53  0.0468  11  56  34  0.0156  
2,000  7  23  8  0  10  49  26  0.0468  13  66  35  0.0468  
1,000  7  26  9  0.0624  4  40  28  0.156  12  124  101  0.7332  
30  500  287  452  387  14.6017  365  691  649  21.0913  249  380  334  10.1089 
1,000  292  456  407  44.5071  343  646  593  69.3736  257  404  359  40.8411  
2,000  343  524  477  154.3786  376  721  655  213.9242  271  457  376  122.1176  
1,000  310  474  424  640.2749  376  729  663  1006.2533  236  412  337  512.9157  
31  500  70  110  87  0.3276  73  158  113  0.39  58  115  90  0.2964 
1,000  49  88  66  0.468  50  131  93  0.624  30  68  45  0.312  
2,000  38  69  49  0.7644  64  142  103  1.3416  34  75  49  0.6396  
1,000  27  52  32  2.0904  53  128  90  6.1152  34  85  60  4.0248  
32  500  6  50  9  0.0156  53  144  89  0.0624  4  60  6  0 
1,000  30  88  44  0.0468  18  211  35  0.0468  6  93  7  0.0156  
2,000  3  23  6  0.0156  10  107  16  0.0468  4  45  5  0.0156  
1,000  5  39  10  0.0936  8  80  17  0.1872  4  55  8  0.0936  
33  500  320  442  406  7.6596  427  822  614  11.8405  4  24  7  0.1248 
1,000  268  381  342  20.6233  338  652  483  29.1566  4  25  7  0.39  
2,000  195  269  248  47.1435  235  461  342  66.7996  4  25  7  1.2792  
1,000  145  198  182  137.4681  223  434  326  247.1524  4  25  7  4.5396  
34  500  105  187  146  0.1404  101  234  162  0.156  98  191  143  0.1404 
1,000  33  76  49  0.1248  73  197  136  0.2496  97  203  156  0.2028  
2,000  123  226  172  0.39  173  416  306  0.6708  89  196  141  0.312  
1,000  175  311  243  2.6676  236  537  395  4.6176  151  319  239  2.6676  
35  500  12  25  14  0.0468  13  35  23  0.078  5  18  5  0.0156 
1,000  12  25  14  0.078  11  30  19  0.1248  5  18  5  0.0312  
2,000  11  23  13  0.1716  11  34  23  0.2808  5  18  5  0.0624  
1,000  10  23  14  0.9984  10  31  22  1.56  5  18  5  0.2808  
36  500  37  228  122  0.6864  10  78  17  0.1404  9  34  16  0.078 
1,000  17  75  25  0.2652  11  78  20  0.234  9  51  13  0.156  
2,000  37  137  57  1.1076  11  101  21  0.5148  15  90  34  0.6864  
1,000  16  82  23  2.496  10  82  18  2.1684  15  76  21  2.106  
37  500  11  25  16  0.0156  10  33  19  0  3  17  6  0 
1,000  12  26  15  0.0156  11  29  19  0.0156  3  17  6  0  
2,000  10  22  12  0.0312  10  27  18  0.0312  3  18  6  0.0156  
5,000  8  18  10  0.0936  10  27  18  0.156  3  18  6  0.0624  
38  500  6  75  50  0.0156  10  123  72  0.0624  8  105  51  0.0312 
1,000  4  63  22  0.0156  7  107  43  0.0468  4  56  24  0.0156  
2,000  7  121  65  0.0624  13  211  130  0.1248  8  132  68  0.078  
5,000  5  95  49  0.2184  5  84  34  0.156  6  115  51  0.2496  
39  500  8  127  51  2.7924  10  168  76  3.7596  9  127  62  3.042 
1,000  7  107  40  8.5021  11  188  85  15.7405  9  144  61  11.9341  
2,000  10  142  67  46.7691  16  248  104  81.2453  113  573  189  182.7708  
5,000  9  165  72  698.6037  12  162  78  675.7495  13  181  83  756.8077  
40  500  170  257  218  0.1092  305  578  430  0.1716  105  171  140  0.1404 
1,000  153  231  196  0.0624  265  505  367  0.1404  114  208  155  0.0624  
2,000  147  222  190  0.0936  272  522  390  0.1872  117  188  155  0.078  
5,000  144  233  184  0.3588  305  579  425  0.8424  125  205  172  0.3276 
Declarations
Acknowledgements
This work is supported in part by the NNSF (11171003) of China.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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