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September  2020, 16(5): 2531-2549. doi: 10.3934/jimo.2019068

Optimal switching signal design with a cost on switching action

Wei Xu 1, , Liying Yu 1, , Gui-Hua Lin 1,, and  Zhi Guo Feng 2,3,

1. 

School of Management, Shanghai University, Shanghai, China
2. 

Faculty of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, China
3. 

College of Mathematics Science, Chongqing Normal University, Chongqing, China

* Corresponding author: Gui-Hua Lin

Received  October 2018 Revised  January 2019 Published  September 2020 Early access  July 2019

In this paper, we consider a particular class of optimal switching problem for the linear-quadratic switched system in discrete time, where an optimal switching sequence is designed to minimize the quadratic performance index of the system with a switching cost. This is a challenging issue and studied only by few papers. First, we introduce a total variation function with respect to the switching sequence to measure the volatile switching action. In order to restrain the switching magnitude, it is added to the cost functional as a penalty. Then, the particular optimal switching problem is formulated. With the positive semi-definiteness of matrices, we construct a series of exact lower bounds of the cost functional at each time and the branch and bound method is applied to search all global optimal solutions. For the comparison between different global optimization methods, some numerical examples are given to show the efficiency of our proposed method.

Keywords: Global optimal solution, optimal switching problem, switching cost, branch and bound.
Mathematics Subject Classification: 49M25.
Citation: Wei Xu, Liying Yu, Gui-Hua Lin, Zhi Guo Feng. Optimal switching signal design with a cost on switching action. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2531-2549. doi: 10.3934/jimo.2019068
References:
[1]

H. Axelsson, Y. Wardi, M. Egerstedt and E. I. Verriest, Gradient descent approach to optiomal mode scheduling in hybrid dynamical systems, Journal of Optimization Theory and Applications, 136 (2008), 167-186.  doi: 10.1007/s10957-007-9305-y.  Google Scholar

[2]

S. C. Bengea and R. A. DeCarlo, Optimal control of switching systems, Automatica, 41 (2005), 11-27.  doi: 10.1016/j.automatica.2004.08.003.  Google Scholar

[3]

T. M. Caldwell and T. D. Murphey, Projection-based iterative mode scheduling for switched systems, Nonlinear Analysis: Hybrid Systems, 21 (2016), 59-83.  doi: 10.1016/j.nahs.2015.11.002.  Google Scholar

[4]

Z. G. Feng, K. L. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica, 44 (2008), 1295-1303.  doi: 10.1016/j.automatica.2007.09.024.  Google Scholar

[5]

Z. G. Feng, K. L. Teo and V. Rehbock, Optimal sensor scheduling in continuous time, Dynamic Systems and Applications, 17 (2008), 331-350.   Google Scholar

[6]

Z. G. Feng, K. L. Teo and V. Rehbock, A discrete filled function method for the optimal control of switched systems in discrete time, Optimal Control Applications and Methods, 30 (2009), 585-593.  doi: 10.1002/oca.885.  Google Scholar

[7]

Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512.  doi: 10.3934/jimo.2005.1.499.  Google Scholar

[8]

J. Gao and D. Li, Linear-quadratic switching control with switching cost, Automatica, 48 (2012), 1138-1143.  doi: 10.1016/j.automatica.2012.03.006.  Google Scholar

[9]

Z. Gong, C. Liu and Y. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198.  doi: 10.3934/jimo.2017042.  Google Scholar

[10]

D. Görges, M. Izák and S. Liu, Optimal control and scheduling of switched systems, IEEE Transactions on Automatic Control, 56 (2011), 135-140.  doi: 10.1109/TAC.2010.2085573.  Google Scholar

[11]

J. F. He, W. Xu, Z. G. Feng and X. Yang, On the global optimal solution for linear quadratic problems of switched system, Journal of Industrial and Management Optimization, 15 (2019), 817-832.   Google Scholar

[12]

M. Kamgarpour and C. Tomlin, On optimal control of non-autonomous switched systems with a fixed mode sequence, Journal of Global Optimization, 48 (2012), 1177-1181.  doi: 10.1016/j.automatica.2012.03.019.  Google Scholar

[13]

B. Li and Y. Rong, Joint transceiver optimization for wireless information and energy transfer in nonregenerative MIMO relay systems, IEEE Transactions on Vehicular Technology, 67 (2018), 8348-8362.  doi: 10.1109/TVT.2018.2846556.  Google Scholar

[14]

B. Li and Y. Rong, AF MIMO relay systems with wireless powered relay node and direct link, IEEE Transactions on Communications, 66 (2018), 1508-1519.  doi: 10.1109/TCOMM.2017.2788006.  Google Scholar

[15]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Transactions on Wireless Communications, 16 (2017), 464-474.  doi: 10.1109/TWC.2016.2625246.  Google Scholar

[16]

R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control parametrization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403.  doi: 10.1016/j.mcm.2005.08.012.  Google Scholar

[17]

C. Liu, Z. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006.  Google Scholar

[18]

R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control, Automatica, 49 (2013), 2652-2664.  doi: 10.1016/j.automatica.2013.05.027.  Google Scholar

[19]

R. C. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980.  doi: 10.1016/j.automatica.2008.10.031.  Google Scholar

[20]

W. Lu, P. Zhu and S. Ferrari, A hybrid-adaptive dynamic programming approach for the model-free control of nonlinear switched systems, IEEE Transactions on Automatic Control, 61 (2016), 3203-3208.  doi: 10.1109/TAC.2015.2509421.  Google Scholar

[21]

C. Seatzu, D. Corona, A. Giua and A. Bemporad, Optimal control of continuous-time switched affine systems, IEEE Transactions on Automatic Control, 51 (2006), 726-741.  doi: 10.1109/TAC.2006.875053.  Google Scholar

[22]

Y. Wardi, M. Egerstedt and M. Hale, Switched-mode systems: Gradient-descent algorithms with Armijo step sizes, Discrete Event Dynamic Systems, 25 (2015), 571-599.  doi: 10.1007/s10626-014-0198-2.  Google Scholar

[23]

W. Xu, Z. G. Feng, J. W. Peng and K. F. C. Yiu, Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193.  doi: 10.1016/j.automatica.2016.12.002.  Google Scholar

[24]

W. Xu, Z. G. Feng, G. H. Lin and L. Yu, Optimal scheduling of discrete-time switched linear systems, IMA Journal of Mathematical Control and Information, (2018).  doi: 10.1093/imamci/dny034.  Google Scholar

[25]

H. Yan, Y. Sun and Y. Zhu, A linear-quadratic control problem of uncertain discrete-time switched systems, Journal of Industrial and Management Optimization, 13 (2017), 267-282.  doi: 10.3934/jimo.2016016.  Google Scholar

[26]

F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, Visual miser: an efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.  Google Scholar

[27]

J. Zhai, T. Niu, J. Ye and E. Feng, Optimal control of nonlinear switched system with mixed constraints and its parallel optimization algorithm, Nonlinear Analysis: Hybrid Systems, 25 (2017), 21-40.  doi: 10.1016/j.nahs.2017.02.001.  Google Scholar

show all references

References:
[1]

H. Axelsson, Y. Wardi, M. Egerstedt and E. I. Verriest, Gradient descent approach to optiomal mode scheduling in hybrid dynamical systems, Journal of Optimization Theory and Applications, 136 (2008), 167-186.  doi: 10.1007/s10957-007-9305-y.  Google Scholar

[2]

S. C. Bengea and R. A. DeCarlo, Optimal control of switching systems, Automatica, 41 (2005), 11-27.  doi: 10.1016/j.automatica.2004.08.003.  Google Scholar

[3]

T. M. Caldwell and T. D. Murphey, Projection-based iterative mode scheduling for switched systems, Nonlinear Analysis: Hybrid Systems, 21 (2016), 59-83.  doi: 10.1016/j.nahs.2015.11.002.  Google Scholar

[4]

Z. G. Feng, K. L. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica, 44 (2008), 1295-1303.  doi: 10.1016/j.automatica.2007.09.024.  Google Scholar

[5]

Z. G. Feng, K. L. Teo and V. Rehbock, Optimal sensor scheduling in continuous time, Dynamic Systems and Applications, 17 (2008), 331-350.   Google Scholar

[6]

Z. G. Feng, K. L. Teo and V. Rehbock, A discrete filled function method for the optimal control of switched systems in discrete time, Optimal Control Applications and Methods, 30 (2009), 585-593.  doi: 10.1002/oca.885.  Google Scholar

[7]

Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512.  doi: 10.3934/jimo.2005.1.499.  Google Scholar

[8]

J. Gao and D. Li, Linear-quadratic switching control with switching cost, Automatica, 48 (2012), 1138-1143.  doi: 10.1016/j.automatica.2012.03.006.  Google Scholar

[9]

Z. Gong, C. Liu and Y. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198.  doi: 10.3934/jimo.2017042.  Google Scholar

[10]

D. Görges, M. Izák and S. Liu, Optimal control and scheduling of switched systems, IEEE Transactions on Automatic Control, 56 (2011), 135-140.  doi: 10.1109/TAC.2010.2085573.  Google Scholar

[11]

J. F. He, W. Xu, Z. G. Feng and X. Yang, On the global optimal solution for linear quadratic problems of switched system, Journal of Industrial and Management Optimization, 15 (2019), 817-832.   Google Scholar

[12]

M. Kamgarpour and C. Tomlin, On optimal control of non-autonomous switched systems with a fixed mode sequence, Journal of Global Optimization, 48 (2012), 1177-1181.  doi: 10.1016/j.automatica.2012.03.019.  Google Scholar

[13]

B. Li and Y. Rong, Joint transceiver optimization for wireless information and energy transfer in nonregenerative MIMO relay systems, IEEE Transactions on Vehicular Technology, 67 (2018), 8348-8362.  doi: 10.1109/TVT.2018.2846556.  Google Scholar

[14]

B. Li and Y. Rong, AF MIMO relay systems with wireless powered relay node and direct link, IEEE Transactions on Communications, 66 (2018), 1508-1519.  doi: 10.1109/TCOMM.2017.2788006.  Google Scholar

[15]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Transactions on Wireless Communications, 16 (2017), 464-474.  doi: 10.1109/TWC.2016.2625246.  Google Scholar

[16]

R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control parametrization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403.  doi: 10.1016/j.mcm.2005.08.012.  Google Scholar

[17]

C. Liu, Z. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006.  Google Scholar

[18]

R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control, Automatica, 49 (2013), 2652-2664.  doi: 10.1016/j.automatica.2013.05.027.  Google Scholar

[19]

R. C. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980.  doi: 10.1016/j.automatica.2008.10.031.  Google Scholar

[20]

W. Lu, P. Zhu and S. Ferrari, A hybrid-adaptive dynamic programming approach for the model-free control of nonlinear switched systems, IEEE Transactions on Automatic Control, 61 (2016), 3203-3208.  doi: 10.1109/TAC.2015.2509421.  Google Scholar

[21]

C. Seatzu, D. Corona, A. Giua and A. Bemporad, Optimal control of continuous-time switched affine systems, IEEE Transactions on Automatic Control, 51 (2006), 726-741.  doi: 10.1109/TAC.2006.875053.  Google Scholar

[22]

Y. Wardi, M. Egerstedt and M. Hale, Switched-mode systems: Gradient-descent algorithms with Armijo step sizes, Discrete Event Dynamic Systems, 25 (2015), 571-599.  doi: 10.1007/s10626-014-0198-2.  Google Scholar

[23]

W. Xu, Z. G. Feng, J. W. Peng and K. F. C. Yiu, Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193.  doi: 10.1016/j.automatica.2016.12.002.  Google Scholar

[24]

W. Xu, Z. G. Feng, G. H. Lin and L. Yu, Optimal scheduling of discrete-time switched linear systems, IMA Journal of Mathematical Control and Information, (2018).  doi: 10.1093/imamci/dny034.  Google Scholar

[25]

H. Yan, Y. Sun and Y. Zhu, A linear-quadratic control problem of uncertain discrete-time switched systems, Journal of Industrial and Management Optimization, 13 (2017), 267-282.  doi: 10.3934/jimo.2016016.  Google Scholar

[26]

F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, Visual miser: an efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.  Google Scholar

[27]

J. Zhai, T. Niu, J. Ye and E. Feng, Optimal control of nonlinear switched system with mixed constraints and its parallel optimization algorithm, Nonlinear Analysis: Hybrid Systems, 25 (2017), 21-40.  doi: 10.1016/j.nahs.2017.02.001.  Google Scholar

Figure 1.  Global optimal switching sequences under different penalty parameters in Example 4.1
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Figure 2.  Global optimal switching sequences under different penalty parameters in Example 4.2
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Table 1.  Global optimal solutions under different penalty parameters in Example 4.1
$ \alpha $ Global optimal solution $ \sigma^{\ast} $ Switching times Performance index Switching cost Optimal functional value $ J^{\ast} $
$ 0 $ (1 4 2 1 3 1 4 2 3 2) 9 40 0 40
(1 4 2 1 3 1 4 2 1 1) 8 40 0 40
$ 0.5 $ (1 4 2 1 3 1 4 2 1 1) 8 40 8 48
(1 1 3 4 4 4 2 1 1 1) 4 45 3 48
$ 1 $ (1 1 3 4 4 4 2 1 1 1) 4 45 6 51
$ 2 $ (2 3 2 2 2 2 1 1 1 1) 3 50 6 56
$ 5 $ (1 1 1 2 2 2 2 2 2 2) 1 59 5 64
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$ \alpha $ Global optimal solution $ \sigma^{\ast} $ Switching times Performance index Switching cost Optimal functional value $ J^{\ast} $
$ 0 $ (1 4 2 1 3 1 4 2 3 2) 9 40 0 40
(1 4 2 1 3 1 4 2 1 1) 8 40 0 40
$ 0.5 $ (1 4 2 1 3 1 4 2 1 1) 8 40 8 48
(1 1 3 4 4 4 2 1 1 1) 4 45 3 48
$ 1 $ (1 1 3 4 4 4 2 1 1 1) 4 45 6 51
$ 2 $ (2 3 2 2 2 2 1 1 1 1) 3 50 6 56
$ 5 $ (1 1 1 2 2 2 2 2 2 2) 1 59 5 64
Table 2.  Two kinds of B&B methods with $ \varepsilon = 10^{-5} $ in Example 4.1
$ \alpha $ Approximate B&B method Exact B&B method
$ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time $ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time
$ 0 $ (1 4 2 1 3 1 4 2 3 2) 40 116 0.0826s (1 4 2 1 3 1 4 2 3 2) 40 128 0.1055s
(1 4 2 1 3 1 4 2 1 1) (1 4 2 1 3 1 4 2 1 1)
$ 0.5 $ (1 4 2 1 3 1 4 2 1 1) 48 144 0.1275s (1 4 2 1 3 1 4 2 1 1) 48 152 0.1310s
(1 1 3 4 4 4 2 1 1 1) (1 1 3 4 4 4 2 1 1 1)
$ 1 $ (1 1 3 4 4 4 2 1 1 1) 51 376 0.2371s (1 1 3 4 4 4 2 1 1 1) 51 392 0.2481s
$ 2 $ (2 3 2 2 2 2 1 1 1 1) 56 440 0.2799s (2 3 2 2 2 2 1 1 1 1) 56 444 0.2806s
$ 5 $ (1 1 1 2 2 2 2 2 2 2) 64 360 0.2280s (1 1 1 2 2 2 2 2 2 2) 64 372 0.2346s
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$ \alpha $ Approximate B&B method Exact B&B method
$ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time $ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time
$ 0 $ (1 4 2 1 3 1 4 2 3 2) 40 116 0.0826s (1 4 2 1 3 1 4 2 3 2) 40 128 0.1055s
(1 4 2 1 3 1 4 2 1 1) (1 4 2 1 3 1 4 2 1 1)
$ 0.5 $ (1 4 2 1 3 1 4 2 1 1) 48 144 0.1275s (1 4 2 1 3 1 4 2 1 1) 48 152 0.1310s
(1 1 3 4 4 4 2 1 1 1) (1 1 3 4 4 4 2 1 1 1)
$ 1 $ (1 1 3 4 4 4 2 1 1 1) 51 376 0.2371s (1 1 3 4 4 4 2 1 1 1) 51 392 0.2481s
$ 2 $ (2 3 2 2 2 2 1 1 1 1) 56 440 0.2799s (2 3 2 2 2 2 1 1 1 1) 56 444 0.2806s
$ 5 $ (1 1 1 2 2 2 2 2 2 2) 64 360 0.2280s (1 1 1 2 2 2 2 2 2 2) 64 372 0.2346s
Table 3.  Two kinds of B&B methods with $ \varepsilon = 1 $ in Example 4.1
$ \alpha $ Approximate B&B method Exact B&B method
$ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time $ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time
0 (1 4 2 1 1 1 1 1 3 1) 55 8 0.0115s (1 4 2 1 3 1 4 2 3 2) 40 2796 10.4855s
(1 4 2 1 3 1 4 2 1 1)
$ 0.5 $ (1 4 2 1 1 1 1 1 3 1) 60 8 0.0115s (1 4 2 1 3 1 4 2 1 1) 48 2948 10.7044s
(1 1 3 4 4 4 2 1 1 1)
$ 1 $ (1 1 3 4 4 4 2 1 1 1) 51 4 0.0091s (1 1 3 4 4 4 2 1 1 1) 51 2768 10.4657s
$ 2 $ (1 1 3 4 4 4 2 1 1 1) 57 12 0.1379s (2 3 2 2 2 2 1 1 1 1) 56 2504 10.3680s
(1 1 3 4 4 4 2 2 2 2)
$ 5 $ (1 1 3 4 4 4 4 3 3 3) 70 4 0.0091s (1 1 1 2 2 2 2 2 2 2) 64 2028 9.6311s
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$ \alpha $ Approximate B&B method Exact B&B method
$ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time $ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time
0 (1 4 2 1 1 1 1 1 3 1) 55 8 0.0115s (1 4 2 1 3 1 4 2 3 2) 40 2796 10.4855s
(1 4 2 1 3 1 4 2 1 1)
$ 0.5 $ (1 4 2 1 1 1 1 1 3 1) 60 8 0.0115s (1 4 2 1 3 1 4 2 1 1) 48 2948 10.7044s
(1 1 3 4 4 4 2 1 1 1)
$ 1 $ (1 1 3 4 4 4 2 1 1 1) 51 4 0.0091s (1 1 3 4 4 4 2 1 1 1) 51 2768 10.4657s
$ 2 $ (1 1 3 4 4 4 2 1 1 1) 57 12 0.1379s (2 3 2 2 2 2 1 1 1 1) 56 2504 10.3680s
(1 1 3 4 4 4 2 2 2 2)
$ 5 $ (1 1 3 4 4 4 4 3 3 3) 70 4 0.0091s (1 1 1 2 2 2 2 2 2 2) 64 2028 9.6311s
Table 4.  Global optimal solutions under different penalty parameters in Example 4.2
$ \alpha $ Global optimal solution $ \sigma^{\ast} $ Switching times Performance index Switching cost Optimal functional value $ J^{\ast} $
0 (3 4 2 3 4 1 1 3 3 4) 7 5.1452 0 5.1452
0.1 (3 4 2 4 1 4 4 4 4 4) 5 5.2376 0.5 5.7376
0.5 (2 3 3 2 4 4 4 4 4 4) 3 5.6759 1.5 7.1759
2 (2 3 3 2 4 4 4 4 4 4) 3 5.6759 6 11.6759
5 (2 3 3 3 4 4 4 4 4 4) 2 7.7973 10 17.7973
10 (4 4 4 4 4 4 4 4 4 4) 0 18.5961 0 18.5961
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$ \alpha $ Global optimal solution $ \sigma^{\ast} $ Switching times Performance index Switching cost Optimal functional value $ J^{\ast} $
0 (3 4 2 3 4 1 1 3 3 4) 7 5.1452 0 5.1452
0.1 (3 4 2 4 1 4 4 4 4 4) 5 5.2376 0.5 5.7376
0.5 (2 3 3 2 4 4 4 4 4 4) 3 5.6759 1.5 7.1759
2 (2 3 3 2 4 4 4 4 4 4) 3 5.6759 6 11.6759
5 (2 3 3 3 4 4 4 4 4 4) 2 7.7973 10 17.7973
10 (4 4 4 4 4 4 4 4 4 4) 0 18.5961 0 18.5961
Table 5.  Two kinds of B&B methods with $ \varepsilon = 10^{-20} $ in Example 4.2
$ \alpha $ Approximate B&B method Exact B&B method
$ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time $ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time
0 (3 4 2 3 4 1 1 3 3 4) 5.1452 164 0.5458s (3 4 2 3 4 1 1 3 3 4) 5.1452 168 0.6742s
0.1 (3 4 2 4 1 4 4 4 4 4) 5.7376 108 0.6379s (3 4 2 4 1 4 4 4 4 4) 5.7376 64 0.7968s
0.5 (2 3 3 2 4 4 4 4 4 4) 7.1759 96 0.7146s (2 3 3 2 4 4 4 4 4 4) 7.1759 120 0.9573s
2 (2 3 3 2 4 4 4 4 4 4) 11.6759 132 1.5675s (2 3 3 2 4 4 4 4 4 4) 11.6759 172 1.7968s
5 (2 3 3 3 4 4 4 4 4 4) 17.7973 36 1.2749s (2 3 3 3 4 4 4 4 4 4) 17.7973 36 1.6238s
10 (4 4 4 4 4 4 4 4 4 4) 18.5961 24 0.8772s (4 4 4 4 4 4 4 4 4 4) 18.5961 68 1.2210s
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$ \alpha $ Approximate B&B method Exact B&B method
$ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time $ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time
0 (3 4 2 3 4 1 1 3 3 4) 5.1452 164 0.5458s (3 4 2 3 4 1 1 3 3 4) 5.1452 168 0.6742s
0.1 (3 4 2 4 1 4 4 4 4 4) 5.7376 108 0.6379s (3 4 2 4 1 4 4 4 4 4) 5.7376 64 0.7968s
0.5 (2 3 3 2 4 4 4 4 4 4) 7.1759 96 0.7146s (2 3 3 2 4 4 4 4 4 4) 7.1759 120 0.9573s
2 (2 3 3 2 4 4 4 4 4 4) 11.6759 132 1.5675s (2 3 3 2 4 4 4 4 4 4) 11.6759 172 1.7968s
5 (2 3 3 3 4 4 4 4 4 4) 17.7973 36 1.2749s (2 3 3 3 4 4 4 4 4 4) 17.7973 36 1.6238s
10 (4 4 4 4 4 4 4 4 4 4) 18.5961 24 0.8772s (4 4 4 4 4 4 4 4 4 4) 18.5961 68 1.2210s
Table 6.  Two kinds of B&B methods with $ \varepsilon = 1 $ in Example 4.2
$ \alpha $ Approximate B&B method Exact B&B method
$ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time $ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time
0 (2 3 3 2 4 1 4 3 4 4) 5.5347 12 0.0679s (3 4 2 3 4 1 1 3 3 4) 5.1452 660 85.3819s
0.1 (2 3 3 2 4 1 4 4 4 4) 6.1436 4 0.0615s (3 4 2 4 1 4 4 4 4 4) 5.7376 568 86.7247s
0.5 (2 3 3 2 4 4 4 4 4 4) 7.1759 4 0.0618s (2 3 3 2 4 4 4 4 4 4) 7.1759 508 87.3288s
2 (2 3 3 2 4 4 4 4 4 4) 11.6759 24 0.1131s (2 3 3 2 4 4 4 4 4 4) 11.6759 408 86.0455s
5 (2 3 3 3 4 4 4 4 4 4) 17.7973 4 0.0797s (2 3 3 3 4 4 4 4 4 4) 17.7973 372 84.8442s
10 (2 3 3 3 4 4 4 4 4 4) 27.7973 4 0.2317s (4 4 4 4 4 4 4 4 4 4) 18.5961 332 83.2846s
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$ \alpha $ Approximate B&B method Exact B&B method
$ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time $ \sigma^{\ast} $ $ J^{\ast} $ Searching times Time
0 (2 3 3 2 4 1 4 3 4 4) 5.5347 12 0.0679s (3 4 2 3 4 1 1 3 3 4) 5.1452 660 85.3819s
0.1 (2 3 3 2 4 1 4 4 4 4) 6.1436 4 0.0615s (3 4 2 4 1 4 4 4 4 4) 5.7376 568 86.7247s
0.5 (2 3 3 2 4 4 4 4 4 4) 7.1759 4 0.0618s (2 3 3 2 4 4 4 4 4 4) 7.1759 508 87.3288s
2 (2 3 3 2 4 4 4 4 4 4) 11.6759 24 0.1131s (2 3 3 2 4 4 4 4 4 4) 11.6759 408 86.0455s
5 (2 3 3 3 4 4 4 4 4 4) 17.7973 4 0.0797s (2 3 3 3 4 4 4 4 4 4) 17.7973 372 84.8442s
10 (2 3 3 3 4 4 4 4 4 4) 27.7973 4 0.2317s (4 4 4 4 4 4 4 4 4 4) 18.5961 332 83.2846s
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