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: November 30, 2019

Revised: April 30, 2020

Early access: August 2020

Published: November 2021

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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:

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Figure 1. Qubit description on the Bloch sphere

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Figure 2. Pressure vessel design problem

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Figure 3. Welded beam design problem

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Figure 4. Tension/compression spring design problem

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Figure 5. Cantilever beam design problem

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Figure 6. Speed reducer design problem

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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