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A new type of quasi-newton updating formulas based on the new quasi-newton equation
Department of Mathematics, College of Computers Sciences and Mathematics, University of Mosul, Iraq |
The quasi-Newton equation is the very foundation of an assortment of the quasi-Newton methods. Therefore, by using the offered alternative equation, we derive the modified BFGS quasi-Newton updating formulas. In this paper, a new y-technique has been introduced to modify the secant equation of the quasi-Newton methods. Prove the global convergence of this algorithm is associated with a line search rule. The numerical results explain that the offered method is effectual for the known test problems.
References:
[1] |
R. Byrd and J. Nocedal,
A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM J. Numer., 26 (1989), 727-739.
doi: 10.1137/0726042. |
[2] |
X. W. Fang, Q. Ni and M. L. Zeng,
A modified quasi-Newton method for nonlinear equation, Journal of Computational and Applied Mathematics, 328 (2018), 44-58.
doi: 10.1016/j.cam.2017.06.024. |
[3] |
R. Fletcher, Practical Methods of Optimization, John Wiley and Sons, ChiChester, New York, 1987. |
[4] |
A. R. M. Issam, A new limited memory Quasi-Newton method for unconstrained optimization, J. KSIAM, 7 (2003), 7-14. Google Scholar |
[5] |
D. Li and M. Fukushima,
A modified BFGS method and its global convergence in nonconvex minimization, J. Comput. Appl. Math., 129 (2001), 15-35.
doi: 10.1016/S0377-0427(00)00540-9. |
[6] |
J. More, B. Garbow and K. Hillstrome,
Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41.
doi: 10.1145/355934.355936. |
[7] |
J. Nocedal and J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer Verlag, New York, USA.
doi: 10.1007/b98874. |
[8] |
M. J. D. Powell,
Algorithms for nonlinear constraints that use Lagrange functions, Math. Programming, 14 (1978), 224-248.
doi: 10.1007/BF01588967. |
[9] |
Z. Wei, G. Li and L. Qi,
New quasi-Newton methods for unconstrained optimization problems, Math Program. Applied Mathematics and Computation, 175 (2006), 1156-1188.
doi: 10.1016/j.amc.2005.08.027. |
[10] |
Z. Wei, G. Li and L. Qi,
The superlinear convergence of a modified BFGS- type method for unconstrained optimization, Comput. Optim. Appl., 29 (2004), 315-332.
doi: 10.1023/B:COAP.0000044184.25410.39. |
[11] |
P. Wolfe,
Convergence conditions for ascent methods, (Ⅱ): Some corrections, SIAM Review, 13 (1971), 185-188.
doi: 10.1137/1013035. |
[12] |
Y. H. Xiao, Z. X. Wei and L. Zhang,
A modified BFGS method without line searches for nonconvex unconstrained optimization, Advances in Theoretical and Applied Mathematics, 1 (2006), 149-162.
|
[13] |
Y. Yuan and W. Sun, Theory and Methods of Optimization, Science Press of China, 1999. |
[14] |
G. Yuan and Z. Wei,
Convergence analysis of a modified BFGS method on convex minimizations, Comp. Optim. Appl., 47 (2010), 237-255.
doi: 10.1007/s10589-008-9219-0. |
[15] |
G. Yuan, Z. Wei and X. Lu,
Global convergence of BFGS and PRP methods under a modified weak Wolfe-Powell line search, Applied Mathematical Modelling, 47 (2017), 811-825.
doi: 10.1016/j.apm.2017.02.008. |
[16] |
G. Yuan, Z. Sheng, B. Wang, W. Hu and C. Li,
The global convergence of a modified BFGS method for nonconvex functions, Journal of Computational and Applied Mathematics, 327 (2018), 274-294.
doi: 10.1016/j.cam.2017.05.030. |
[17] |
G. Yuan, Z. Wei and Y. Wu,
Modified limited memory BFGS method with nonmonotone line search for unconstrained optimization, J. Korean Math. Soc., 47 (2010), 767-788.
doi: 10.4134/JKMS.2010.47.4.767. |
[18] |
J. Z. Zhang, N. Y. Deng and L. H. Chen,
Quasi-Newton equation and related methods for unconstrained optimization, JOTA, 102 (1999), 147-167.
doi: 10.1023/A:1021898630001. |
show all references
References:
[1] |
R. Byrd and J. Nocedal,
A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM J. Numer., 26 (1989), 727-739.
doi: 10.1137/0726042. |
[2] |
X. W. Fang, Q. Ni and M. L. Zeng,
A modified quasi-Newton method for nonlinear equation, Journal of Computational and Applied Mathematics, 328 (2018), 44-58.
doi: 10.1016/j.cam.2017.06.024. |
[3] |
R. Fletcher, Practical Methods of Optimization, John Wiley and Sons, ChiChester, New York, 1987. |
[4] |
A. R. M. Issam, A new limited memory Quasi-Newton method for unconstrained optimization, J. KSIAM, 7 (2003), 7-14. Google Scholar |
[5] |
D. Li and M. Fukushima,
A modified BFGS method and its global convergence in nonconvex minimization, J. Comput. Appl. Math., 129 (2001), 15-35.
doi: 10.1016/S0377-0427(00)00540-9. |
[6] |
J. More, B. Garbow and K. Hillstrome,
Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41.
doi: 10.1145/355934.355936. |
[7] |
J. Nocedal and J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer Verlag, New York, USA.
doi: 10.1007/b98874. |
[8] |
M. J. D. Powell,
Algorithms for nonlinear constraints that use Lagrange functions, Math. Programming, 14 (1978), 224-248.
doi: 10.1007/BF01588967. |
[9] |
Z. Wei, G. Li and L. Qi,
New quasi-Newton methods for unconstrained optimization problems, Math Program. Applied Mathematics and Computation, 175 (2006), 1156-1188.
doi: 10.1016/j.amc.2005.08.027. |
[10] |
Z. Wei, G. Li and L. Qi,
The superlinear convergence of a modified BFGS- type method for unconstrained optimization, Comput. Optim. Appl., 29 (2004), 315-332.
doi: 10.1023/B:COAP.0000044184.25410.39. |
[11] |
P. Wolfe,
Convergence conditions for ascent methods, (Ⅱ): Some corrections, SIAM Review, 13 (1971), 185-188.
doi: 10.1137/1013035. |
[12] |
Y. H. Xiao, Z. X. Wei and L. Zhang,
A modified BFGS method without line searches for nonconvex unconstrained optimization, Advances in Theoretical and Applied Mathematics, 1 (2006), 149-162.
|
[13] |
Y. Yuan and W. Sun, Theory and Methods of Optimization, Science Press of China, 1999. |
[14] |
G. Yuan and Z. Wei,
Convergence analysis of a modified BFGS method on convex minimizations, Comp. Optim. Appl., 47 (2010), 237-255.
doi: 10.1007/s10589-008-9219-0. |
[15] |
G. Yuan, Z. Wei and X. Lu,
Global convergence of BFGS and PRP methods under a modified weak Wolfe-Powell line search, Applied Mathematical Modelling, 47 (2017), 811-825.
doi: 10.1016/j.apm.2017.02.008. |
[16] |
G. Yuan, Z. Sheng, B. Wang, W. Hu and C. Li,
The global convergence of a modified BFGS method for nonconvex functions, Journal of Computational and Applied Mathematics, 327 (2018), 274-294.
doi: 10.1016/j.cam.2017.05.030. |
[17] |
G. Yuan, Z. Wei and Y. Wu,
Modified limited memory BFGS method with nonmonotone line search for unconstrained optimization, J. Korean Math. Soc., 47 (2010), 767-788.
doi: 10.4134/JKMS.2010.47.4.767. |
[18] |
J. Z. Zhang, N. Y. Deng and L. H. Chen,
Quasi-Newton equation and related methods for unconstrained optimization, JOTA, 102 (1999), 147-167.
doi: 10.1023/A:1021898630001. |
Author(s) | QN conditions | Ref. |
Powell | [8] | |
Li and Fukushima | [5] | |
Wei, Li, and Qi | [9] | |
Zhang, Deng, and Chen | [18] | |
Yuan and Wei | [14] | |
Yuan, Wei and Wu | [17] |
Author(s) | QN conditions | Ref. |
Powell | [8] | |
Li and Fukushima | [5] | |
Wei, Li, and Qi | [9] | |
Zhang, Deng, and Chen | [18] | |
Yuan and Wei | [14] | |
Yuan, Wei and Wu | [17] |
P.No. | n | BFGS algorithm | BBFGS with |
BBFGS with |
|||
NI | NF | NI | NF | NI | NF | ||
1 | 2 | 35 | 140 | 36 | 124 | 8 | 29 |
2 | 2 | 9 | 26 | 8 | 23 | 5 | 16 |
3 | 2 | 43 | 166 | 34 | 123 | 3 | 12 |
4 | 2 | 3 | 30 | 3 | 30 | 3 | 30 |
5 | 2 | 15 | 50 | 15 | 48 | 5 | 17 |
6 | 2 | 2 | 27 | 2 | 27 | 2 | 27 |
7 | 3 | 34 | 113 | 26 | 86 | 7 | 20 |
8 | 3 | 16 | 54 | 15 | 51 | 6 | 18 |
9 | 3 | 2 | 4 | 2 | 4 | 2 | 4 |
10 | 3 | 2 | 27 | 2 | 27 | 2 | 27 |
11 | 3 | 2 | 27 | 2 | 27 | 2 | 27 |
12 | 4 | 20 | 60 | 20 | 60 | 5 | 17 |
13 | 4 | 19 | 61 | 24 | 73 | 4 | 13 |
14 | 4 | 21 | 65 | 23 | 72 | 4 | 10 |
15 | 4 | 17 | 54 | 16 | 49 | 5 | 17 |
16 | 5 | 2 | 27 | 2 | 27 | 2 | 27 |
17 | 6 | 25 | 72 | 33 | 101 | 4 | 12 |
18 | 11 | 3 | 31 | 3 | 31 | 3 | 31 |
19 | 20 | 31 | 102 | 33 | 103 | 4 | 13 |
20 | 400 | 64 | 209 | 91 | 297 | 5 | 17 |
21 | 400 | 2 | 27 | 2 | 27 | 2 | 27 |
22 | 200 | 2 | 5 | 2 | 5 | 2 | 5 |
23 | 100 | 2 | 27 | 2 | 27 | 2 | 27 |
24 | 500 | 9 | 33 | 8 | 28 | 10 | 31 |
25 | 500 | 2 | 4 | 2 | 4 | 2 | 4 |
26 | 500 | 6 | 16 | 7 | 19 | 5 | 14 |
27 | 500 | 57 | 281 | 16 | 114 | 5 | 17 |
28 | 500 | 2 | 4 | 2 | 4 | 2 | 4 |
29 | 500 | 3 | 7 | 3 | 7 | 3 | 7 |
30 | 500 | 3 | 7 | 3 | 7 | 3 | 7 |
Total | 453 | 1756 | 437 | 1625 | 117 | 527 |
P.No. | n | BFGS algorithm | BBFGS with |
BBFGS with |
|||
NI | NF | NI | NF | NI | NF | ||
1 | 2 | 35 | 140 | 36 | 124 | 8 | 29 |
2 | 2 | 9 | 26 | 8 | 23 | 5 | 16 |
3 | 2 | 43 | 166 | 34 | 123 | 3 | 12 |
4 | 2 | 3 | 30 | 3 | 30 | 3 | 30 |
5 | 2 | 15 | 50 | 15 | 48 | 5 | 17 |
6 | 2 | 2 | 27 | 2 | 27 | 2 | 27 |
7 | 3 | 34 | 113 | 26 | 86 | 7 | 20 |
8 | 3 | 16 | 54 | 15 | 51 | 6 | 18 |
9 | 3 | 2 | 4 | 2 | 4 | 2 | 4 |
10 | 3 | 2 | 27 | 2 | 27 | 2 | 27 |
11 | 3 | 2 | 27 | 2 | 27 | 2 | 27 |
12 | 4 | 20 | 60 | 20 | 60 | 5 | 17 |
13 | 4 | 19 | 61 | 24 | 73 | 4 | 13 |
14 | 4 | 21 | 65 | 23 | 72 | 4 | 10 |
15 | 4 | 17 | 54 | 16 | 49 | 5 | 17 |
16 | 5 | 2 | 27 | 2 | 27 | 2 | 27 |
17 | 6 | 25 | 72 | 33 | 101 | 4 | 12 |
18 | 11 | 3 | 31 | 3 | 31 | 3 | 31 |
19 | 20 | 31 | 102 | 33 | 103 | 4 | 13 |
20 | 400 | 64 | 209 | 91 | 297 | 5 | 17 |
21 | 400 | 2 | 27 | 2 | 27 | 2 | 27 |
22 | 200 | 2 | 5 | 2 | 5 | 2 | 5 |
23 | 100 | 2 | 27 | 2 | 27 | 2 | 27 |
24 | 500 | 9 | 33 | 8 | 28 | 10 | 31 |
25 | 500 | 2 | 4 | 2 | 4 | 2 | 4 |
26 | 500 | 6 | 16 | 7 | 19 | 5 | 14 |
27 | 500 | 57 | 281 | 16 | 114 | 5 | 17 |
28 | 500 | 2 | 4 | 2 | 4 | 2 | 4 |
29 | 500 | 3 | 7 | 3 | 7 | 3 | 7 |
30 | 500 | 3 | 7 | 3 | 7 | 3 | 7 |
Total | 453 | 1756 | 437 | 1625 | 117 | 527 |
BFGS algorithm | BBFGS with |
BBFGS with |
|
NI | 100% | 96.70% | 25.82% |
NF | 100% | 92.53% | 30.01% |
BFGS algorithm | BBFGS with |
BBFGS with |
|
NI | 100% | 96.70% | 25.82% |
NF | 100% | 92.53% | 30.01% |
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