Quantum-inspired satin bowerbird algorithm with Bloch spherical search for constrained structural optimization

Sen Zhang; Guo Zhou; Yongquan Zhou; Qifang Luo

doi:10.3934/jimo.2020130

Article Contents

2021, Volume 17, Issue 6: 3509-3523. Doi: 10.3934/jimo.2020130

This issue Previous Article A polyhedral conic functions based classification method for noisy data Next Article Continuity, differentiability and semismoothness of generalized tensor functions

Quantum-inspired satin bowerbird algorithm with Bloch spherical search for constrained structural optimization

1.
College of Information Science and Engineering, Guangxi University for Nationalities, Nanning 530006, China
2.
Department of Science and Technology Teaching, China University of Political Science and Law, Beijing 102249, China
3.
Guangxi Key Laboratories of Hybrid Computation and IC Design Analysis, Key Laboratory of Guangxi High Schools Complex System and Computational Intelligence, Nanning 530006, China

^* Corresponding author: Guo Zhou
^* Corresponding author: Guo Zhou

Received: December 2019

Revised: May 2020

Early access: August 2020

Published: November 2021

Abstract Full Text(HTML) Figure(6) / Table(10) Related Papers Cited by

Abstract

To enhance the optimization ability of the satin bowerbird optimization (SBO) algorithm, in this paper, a novel quantum-inspired SBO with Bloch spherical search is proposed. In this algorithm, satin bowerbirds are encoded using qubits described on the Bloch sphere, each satin bowerbird occupies three locations in the search space and each location represents an optimization solution. Using the search method of general SBO to adjust the two parameters of the qubit, qubit rotation is performed on the Bloch sphere, which simultaneously updates the three locations occupied by a qubit and quickly approaches the global optimal solution. Finally, the experimental results of five examples of structural engineering design show that the proposed algorithm is superior to other state-of-the-art metaheuristic algorithms in terms of the performance measures.

Keywords:

Quantum coding Bloch spherical coordinates,
quantum-inspired satin bowerbird algorithm,
constrained structural optimization,
metaheuristic.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

Figure 1. Qubit description on the Bloch sphere

Download: Full-size image PowerPoint slide

Figure 2. Pressure vessel design problem

Download: Full-size image PowerPoint slide

Figure 3. Welded beam design problem

Download: Full-size image PowerPoint slide

Figure 4. Tension/compression spring design problem

Download: Full-size image PowerPoint slide

Figure 5. Cantilever beam design problem

Download: Full-size image PowerPoint slide

Figure 6. Speed reducer design problem

Download: Full-size image PowerPoint slide

Table 1. Comparison results for the pressure vessel design problem with optimal

Algorithms	$T_s$	$T_h$	$R$	$L$	Optimal cost
GSA [37]	1.125	0.625	55.988	84.454	8538.835
PSO (He and Wang) [16]	0.812	0.437	42.091	176.74	6061.077
GA (Coello) [27]	0.812	0.434	40.323	200.00	6288.7445
GA(Coello and Montes) [6]	0.812	0.437	42.097	176.654	6059.94
GA (Deb and Gene) [9]	0.937	0.500	48.329	112.679	6410.38
ES(Montes and Coello) [29]	0.812	0.437	42.098	176.640	6059.74
DE (Huang et al.) [24]	0.812	0.437	42.098	176.637	6059.73
ACO(Kaveh and Talataheri) [20]	0.812	0.437	42.103	176.572	6059.08
Lagrangian Multiplier (Kannan) [18]	1.125	0.625	58.291	43.6900	7198.04
Branch-bound (Sandgren) [38]	1.125	0.625	47.700	117.701	8129.10
SBO [33]	0.940	0.468	46.088	116.438	6176.02
D-DS [26]	0.8125	0.4375	42.098	176.637	6059.71
QBSBO	0.835	0.411	43.082	164.816	5998.92

| Show Table

DownLoad: CSV

Table 2. Comparison statistical results for the pressure vessel design problem

Algorithms	Best	Worst	Average	Std
D-DS [26]	6059.71	6410.02	6121.42	23.81
SBO [33]	6176.02	7113.5	6792.53	257.96
QBSBO	5998.67	7069.8	6434.56	189.75

| Show Table

DownLoad: CSV

Table 3. Comparison results for the pressure vessel design problem with optimal

Algorithms	$h$	$l$	$t$	$b$	Optimal cost
GSA [37]	0.1821	3.8569	10.00	0.202	1.879
CPSO [22]	0.202	3.5442	9.0482	0.2057	1.728
GA(Coello) [5]	N/A	N/A	N/A	N/A	1.824
GA(Deb) [7]	N/A	N/A	N/A	N/A	2.380
GA(Deb) [8]	0.248	6.173	8.1789	0.2533	2.433
HS(Leeand Geem) [23]	0.244	6.223	8.2915	0.2443	2.380
Random [34]	0.457	4.731	5.085	0.660	4.118
Simplex [34]	0.279	5.625	7.751	0.279	2.530
David [34]	0.243	6.255	8.291	0.244	2.384
Approx [34]	0.244	6.218	8.291	0.244	2.381
SBO [33]	0.214	3.492	8.557	0.229	1.849
D-DS [26]	0.206	3.253	9.037	0.206	1.696
QBSBO	0.213	3.519	8.492	0.233	1.826

| Show Table

DownLoad: CSV

Table 4. Comparison statistical results for the welded beam design problem

Algorithms	Best	Worst	Average	Std
D-DS [26]	1.695	1.695	1.695	1.39e-06
SBO	1.849	3.046	2.532	0.4280
QBSBO	1.826	2.153	1.903	0.1354

| Show Table

DownLoad: CSV

Table 5. Comparison results for the compression spring design problem

Algorithms	$d$	$D$	$N$	Optimal cost
GSA [37]	0.0502	0.3236	13.525	0.0127
PSO(Ha and Wang) [16]	0.0517	0.3576	11.244	0.01267
ES(Coello and Montes) [29]	0.0519	0.3639	10.890	0.01268
GA (Coello) [27]	0.0514	0.3516	11.632	0.01270
Montes and Coello [24]	0.0516	0.3553	11.397	0.0126
Constraintcorrection(Arora) [20]	0.0500	0.3159	14.250	0.0128
Mathematical optimization(Belegundu) [2]	0.0533	0.3991	9.1854	0.0127
SBO [33]	0.055	0.464	7.0393	0.0131
D-DS [26]	0.052	0.356	11.3434	0.0127
QBSBO	0.051	0.357	11.28	0.0127

| Show Table

DownLoad: CSV

Table 6. Compression spring design problem

Algorithms	Best	Worst	Average	Std
D-DS [26]	0.0127	0.0127	0.0127	0.00001
SBO	0.0131	0.0183	0.0149	0.0014
QBSBO	0.0127	0.0150	0.0134	0.0005

| Show Table

DownLoad: CSV

Table 7. Cantilever design problem

Algorithms	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	Optimal cost
MMA [4]	6.0100	5.3000	4.4900	3.4900	2.1500	1.3400
GCA_I [4]	6.0100	5.3000	4.4900	3.4900	2.1500	1.3400
GCA_II [4]	6.0100	5.3000	4.4900	3.4900	2.1500	1.3400
CS [31]	6.0089	5.3049	4.5023	3.5077	2.1504	1.3399
SOS [3]	6.0187	5.3034	4.4958	3.4989	2.1556	1.3399
SBO	6.1900	5.3221	4.3851	3.4203	2.1761	1.3412
QBSBO	6.0091	5.2999	4.4901	3.4789	2.1531	1.3400

| Show Table

DownLoad: CSV

Table 8. Cantilever design problem

Algorithms	Best	Worst	Average	Std
SBO	1.3514	1.3564	1.3468	0.00378
QBSBO	1.3400	1.3536	1.3446	0.00409

| Show Table

DownLoad: CSV

Table 9. Comparison results for the speed reducer design problem

Algorithms	$b$	$m$	$z$	$l_1$	$l_2$	$d_1$	$d_2$	Optimal cost
Akhtar et al. [1]	3.5061	0.7000	17	7.5491	7.8593	3.3655	5.28977	3008.08
Mezura-Montes et al. [30]	3.5061	0.7008	17	7.1601	7.9621	3.3629	5.3090	3025.005
CS [31]	3.5015	0.7	17	7.6050	7.8181	3.3520	5.2875	3000.981
HCPS [28]	3.5	0.7	17	7.3	7.7153	3.3502	5.28665	2994.471
SCA [36]	3.5	0.7	17	7.327602	7.7153	3.3502	5.2866	2994.744
ABC [35]	3.499999	0.7	17	7.3	7.8	3.3502	5.2878	2997.058
SBO	3.5036	0.7000	17	7.5376	7.3000	3.3579	5.2935	2998.9
QBSBO	3.50000	0.70000	17	7.30000	7.30000	3.3502	5.28652	2985.142

| Show Table

DownLoad: CSV

Table 10. Comparison results for the speed reducer design problem

Algorithms	Best	Worst	Average	Std
SBO	2998.9	3108.540	3065.71	23.017
QBSBO	2985.185	3366.970	3084.75	144.36

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	S. Akhtar, K. Tai and T. Ray, A socio-behavioural simulation model for engineering design optimization, Engineering Optimization, 34 (2002), 341-354. doi: 10.1080/03052150212723.
[2]	A. D. Belegundu and J. S. Arora, A study of mathematical programming methods for structural optimization. Part I: Theory, International Journal for Numerical Methods in Engineering, 21 (1985), 1583-1599. doi: 10.1002/nme.1620210904.
[3]	M. Y. Cheng and D. Prayogo, Symbiotic organisms search: a new metaheuristic optimization algorithm, Computers & Structures, 139 (2014), 98-112. doi: 10.1016/j.compstruc.2014.03.007.
[4]	H. Chickermane and H. C. Gea, Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39 (1996), 829-846. doi: 10.1002/(SICI)1097-0207(19960315)39:5<829::AID-NME884>3.0.CO;2-U.
[5]	C. A. C. Coello, Constraint-handling using an evolutionary multiobjective optimization technique, Civil Engineering Systems, 17 (2000), 319-346. doi: 10.1080/02630250008970288.
[6]	C. A. C. Coello and E. M. Montes, Constraint-handling in genetic algorithms through the use of dominance-based tournament selection, Advanced Engineering Informatics, 16 (2002), 193-203. doi: 10.1016/S1474-0346(02)00011-3.
[7]	K. Deb, Optimal design of a welded beam via genetic algorithms, AIAA Journal, 29 (1991), 2013-2015. doi: 10.2514/3.10834.
[8]	K. Deb, An efficient constraint handling method for genetic algorithms, Computer Methods in Applied Mechanics and Engineering, 186 (2000), 311-338. doi: 10.1016/S0045-7825(99)00389-8.
[9]	K. Deb, GeneAS: A robust optimal design technique for mechanical component design, in Evolutionary Algorithms in Engineering Applications, Springer, Berlin, Heidelberg, 1997,497–514. doi: 10.1007/978-3-662-03423-1_27.
[10]	K. Deb, A. Pratap and S. Agarwal, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197. doi: 10.1109/4235.996017.
[11]	D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proceedings of the Royal Society of London Series A, 400 (1985), 97-117. doi: 10.1098/rspa.1985.0070.
[12]	M. Dorigo, M. Birattari and T. Stutzle, Ant colony optimization, IEEE Computational Intelligence Magazine, (2006), 28–39.
[13]	A. H. Gandomi and X. S. Yang, Benchmark problems in structural optimization, in Computational Optimization, Methods and Algorithms, Springer, Berlin, 2011,259–281. doi: 10.1007/978-3-642-20859-1_12.
[14]	A. H. Gandomi, X. S. Yang and A. H. Alavi, Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems, Engineering with Computers, 29 (2013), 17-35.
[15]	K. H. Han and J. H. Kim, Genetic quantum algorithm and its application to combinatorial optimization problem, Proceedings of the 2000 Congress on Evolutionary Computation, 2 (2000), 1354-1360. doi: 10.1109/CEC.2000.870809.
[16]	Q. He and L. Wang, An effective co-evolutionary particle swarm optimization for constrained engineering design problems, Engineering Applications of Artificial Intelligence, 20 (2007), 89-99. doi: 10.1016/j.engappai.2006.03.003.
[17]	C. Hui, Z. Jiashu and Z. Chao, Chaos updating rotated gates quantum-inspired genetic algorithm. Communications, Circuits and Systems, 2004 International Conference on Communications, Circuits and Systems, Chengdu, 2 (2004), 1108–1112. doi: 10.1109/ICCCAS.2004.1346370.
[18]	B. K. Kannan and S. N. Kramer, An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design, Journal of Mechanical Design, 116 (1994), 405-411. doi: 10.1115/1.2919393.
[19]	D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm, Journal of Global Optimization, 2007, 39(4), 459–471. doi: 10.1007/s10898-007-9149-x.
[20]	A. Kaveh and S. Talatahari, An improved ant colony optimization for constrained engineering design problems, Engineering Computations, 27 (2010), 155-182. doi: 10.1108/02644401011008577.
[21]	J. Kennedy, Particle Swarm Optimization. Encyclopedia of Machine Learning, Springer, Boston, MA, 2011,760–766. doi: 10.1007/978-0-387-30164-8_630.
[22]	R. A. Krohling and L. dos Santos Coelho, Coevolutionary particle swarm optimization using Gaussian distribution for solving constrained optimization problems, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 36 (2006), 1407-1416. doi: 10.1109/TSMCB.2006.873185.
[23]	K. S. Lee and Z. W. Geem, A new meta-heuristic algorithm for continuous engineering optimization: Harmony search theory and practice, Computer Methods in Applied Mechanics and Engineering, 194 (2005), 3902-3933. doi: 10.1016/j.cma.2004.09.007.
[24]	L. J. Li, Z. B. Huang and F. Liu, A heuristic particle swarm optimizer for optimization of pin connected structures, Computers & Structures, 85 (2007), 340-349. doi: 10.1016/j.compstruc.2006.11.020.
[25]	P. Li and S. Li, Quantum-inspired evolutionary algorithm for continuous space optimization based on Bloch coordinates of qubits, Neurocomputing, 72 (2008), 581-591. doi: 10.1016/j.neucom.2007.11.017.
[26]	J. Liu, C. Wu, G. Wu and X. Wang, A novel differential search algorithm and applications for structure design, Applied Mathematics and Computation, 268 (2015), 246-269. doi: 10.1016/j.amc.2015.06.036.
[27]	F. S. Lobato, V. Steffen and Jr ., Fish swarm optimization algorithm applied to engineering system design, Latin American Journal of Solids and Structures, 11 (2014), 143-156.
[28]	W. Long, W. Zhang, Y. Huang and Y. Chen, A hybrid cuckoo search algorithm with feasibility-based rule for constrained structural optimization, Journal of Central South University, 21 (2014), 3197-3204. doi: 10.1007/s11771-014-2291-y.
[29]	E. Mezura-Montes and C. A. C. Coello, An empirical study about the usefulness of evolution strategies to solve constrained optimization problems, International Journal of General Systems, 37 (2008), 443-473. doi: 10.1080/03081070701303470.
[30]	E. Mezura-Montes, C. A. C. Coello and R. Landa-Becerra, Engineering optimization using simple evolutionary algorithm, in Proceedings. 15th IEEE International Conference on Tools with Artificial Intelligence, Sacramento, CA, 2003,149–156. doi: 10.1109/TAI.2003.1250183.
[31]	S. Mirjalili, Dragonfly algorithm: A new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems, Neural Computing and Applications, 27 (2016), 1053-1073.
[32]	S. Mirjalili, S. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61.
[33]	S. H. S. Moosavi and V. K. Bardsiri, Satin bowerbird optimizer: A new optimization algorithm to optimize ANFIS for software development effort estimation, Engineering Applications of Artificial Intelligence, 60 (2012), 1-15. doi: 10.1016/j.engappai.2017.01.006.
[34]	K. M. Ragsdell and D. T. Phillips, Optimal design of a class of welded structures using geometric programming, Journal of Manufacturing Science and Engineering, 98 (1976), 1021-1025. doi: 10.1115/1.3438995.
[35]	S. S. Rao, Engineering Optimization: Theory and Practice, John Wiley & Sons, Inc., New York, 2009.
[36]	T. Ray and K. M. Liew, Society and civilization: An optimization algorithm based on the simulation of social behavior, IEEE Transactions on Evolutionary Computation, 7 (2003), 386-396. doi: 10.1109/TEVC.2003.814902.
[37]	E. Rashedi, H. Nezamabadi-Pour and S. Saryazdi, GSA: A gravitational search algorithm, Information Sciences, 179 (2009), 2232-2248.
[38]	E. Sandgren, Nonlinear integer and discrete programming in mechanical design optimization, Journal of Mechanical Design, 112 (1990), 223-229. doi: 10.1115/1.2912596.
[39]	R. Storn and K. Price, Differential evolution – A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359. doi: 10.1023/A:1008202821328.
[40]	L. Wang, F. Tang and H. Wu, Hybrid genetic algorithm based on quantum computing for numerical optimization and parameter estimation, Applied Mathematics and Computation, 171 (2005), 1141-1156. doi: 10.1016/j.amc.2005.01.115.
[41]	D. H. Wolpert and W. G. Macready, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation, 1 (1997), 67-82. doi: 10.1109/4235.585893.
[42]	X. S. Yang, Flower pollination algorithm for global optimization, Unconventional Computation and Natural Computation, Springer, Berlin, Heidelberg, 2012,240–249. doi: 10.1007/978-3-642-32894-7_27.
[43]	G. Zhang, W. Jin and L. Hu, A novel parallel quantum genetic algorithm, Proceedings of the Fourth International Conference on Parallel and Distributed Computing, Applications and Technologies, Chengdu, China, 2003,693–697. doi: 10.1109/PDCAT.2003.1236393.