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A parallel boundary search particle swarm optimization algorithm for constrained optimization problems

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Abstract

During the past decade, considerable research has been conducted on constrained optimization problems (COPs) which are frequently encountered in practical engineering applications. By introducing resource limitations as constraints, the optimal solutions in COPs are generally located on boundaries of feasible design space, which leads to search difficulties when applying conventional optimization algorithms, especially for complex constraint problems. Even though penalty function method has been frequently used for handling the constraints, the adjustment of control parameters is often complicated and involves a trial-and-error approach. To overcome these difficulties, a modified particle swarm optimization (PSO) algorithm named parallel boundary search particle swarm optimization (PBSPSO) algorithm is proposed in this paper. Modified constrained PSO algorithm is adopted to conduct global search in one branch while Subset Constrained Boundary Narrower (SCBN) function and sequential quadratic programming (SQP) are applied to perform local boundary search in another branch. A cooperative mechanism of the two branches has been built in which locations of the particles near boundaries of constraints are selected as initial positions of local boundary search and the solutions of local boundary search will lead the global search direction to boundaries of active constraints. The cooperation behavior of the two branches effectively reinforces the optimization capability of the PSO algorithm. The optimization performance of PBSPSO algorithm is illustrated through 13 CEC06 test functions and 5 common engineering problems. The results are compared with other state-of-the-art algorithms and it is shown that the proposed algorithm possesses a competitive global search capability and is effective for constrained optimization problems in engineering applications.

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Acknowledgements

The research leading to the above results was supported by National Natural Science Foundation of China (Grant No. 11772191), National Science Foundation for Young Scientists of China (Grant No. 51705312) and National Postdoctoral Foundation of China (Grant No. 17Z102060055). The authors also acknowledge the support from the Adjunct Professor position provided by the Shanghai Jiao Tong University to Prof. Wei Chen.

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Authors and Affiliations

  1. The State key laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 DongChuan RD, Shanghai, 200240, People’s Republic of China

    Zhao Liu, Zeyang Li, Ping Zhu & Wei Chen

  2. Department of Mechanical Engineering, Northwestern University, 2145 Sheridan RD Tech B224, Evanston, IL, 60201, USA

    Wei Chen

Authors
  1. Zhao Liu
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  2. Zeyang Li
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  3. Ping Zhu
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  4. Wei Chen
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Corresponding author

Correspondence to Ping Zhu.

Appendices

Appendix 1 Tension/compression spring design problem

$$ f(X)=\left({x}_3+2\right){x}_2{x}_1^2 $$
(Minimize)

subject to:

$$ {\displaystyle \begin{array}{l}\begin{array}{c}{g}_1(X)=1-\frac{x_2^3{x}_3}{7.1785{x}_4}\le 0\\ {}{g}_2(X)=\frac{4{x}_2^2-{x}_1{x}_2}{12.566\left({x}_2{x}_1^3\right)-{x}_1^4}+\frac{1}{5.108{x}_1^2}-1\le 0\\ {}g3(X)=\frac{140.45{x}_1}{x_2^2{x}_3}1\le 0\\ {}{g}_4(X)=\frac{x_2+{x}_1}{x_2^2{x}_3}-1\le 0\end{array}\\ {} with\;0.05\le {x}_1\le 2.0,0.25\le {x}_2\le 1.3,\\ {} and\;2.0\le {x}_3\le 15.0.\end{array}} $$

Appendix 2 Three bar truss design problem

$$ f(X)=\left(2\sqrt{2}{x}_1+{x}_2\right)l $$
(Minimize)

subject to:

$$ {\displaystyle \begin{array}{l}\begin{array}{c}{g}_1(X)=\frac{\sqrt{2}{x}_1+{x}_2}{\sqrt{2}{x}_1^2+2{x}_1{x}_2}P-\sigma \le 0\\ {}{g}_2(X)=\frac{2}{\sqrt{2}{x}_1^2+2{x}_1{x}_2}P-\sigma \le 0\\ {}{g}_3(X)=\frac{1}{\sqrt{2}{x}_2+{x}_1}P-\sigma \le 0\end{array}\\ {}0\le {x}_i\le 1,i=1,2\\ {}l=100 cm,p=2 kN/{cm}^2,\sigma =2 kN/{cm}^2\end{array}} $$

Appendix 3 Welded beam design problem

$$ f(X)=1.10471{x}_1{x}_2^2+0.04811{x}_3{x}_4\left(14+{x}_2\right) $$
(Minimize)

subject to:

$$ {\displaystyle \begin{array}{l}\begin{array}{l}{g}_1(X)=\tau (X)-{\tau}_{\mathrm{max}}\le 0\\ {}{g}_2(X)=\sigma (X)-{\sigma}_{\mathrm{max}}\le 0\\ {}{g}_3(X)={x}_1-{x}_4\le 0\\ {}{g}_4(X)={0.10471}_1^2+0.4811{x}_3{x}_4\left(14+{x}_2\right)-5\le 0\\ {}{g}_5(X)=0.125-{x}_1\le 0\\ {}{g}_6(X)=\delta (X)-0.25\le 0\\ {}{g}_7(X)=P-{P}_c(X)\le 0\end{array}\\ {}0.1\le {x}_1,{x}_4\le 2\\ {}0.1\le {x}_2,{x}_3\le 10\\ {} where\;\tau (X)=\sqrt{{\tau^{\prime}}^2+2{\tau}^{\prime }{x}_2+2R+{\tau^{{\prime\prime}}}^2}\\ {}{\tau}^{\prime }=\frac{P}{\sqrt{2}{x}_1{x}_2},{\tau}^{{\prime\prime} }=\frac{MR}{J},M=P\left(14\frac{x_2}{2}\right)\\ {}R=\sqrt{\frac{x_2^2}{4}+{\left(\frac{x1+x3}{2}\right)}^2,J=2\left\{\sqrt{2}{x}_1{x}_2\left[\frac{x_2^2}{12}+\left(\frac{x_1+{x}_3}{2}\right)\right]\right\}},\\ {}\sigma (X)=\frac{504000}{x_4{x}_3^2},\delta (X)=\frac{65856000}{\left(30\times {10}^6{x}_4{x}_3^3\right)}\\ {} Pc=\frac{4.013\times \left(30\times {10}^6\right)}{196}\sqrt{\frac{x_3^2{x}_4^6}{36}\times \left[1-\left({x}_3\sqrt{\frac{30\times {10}^6}{4\times \left(12\times {10}^6\right)}/28}\right)\right],}\\ {}P=6000 lb,L=14 in,E=30\times {10}^6 psi,G=12\times {10}^6 psi,\\ {}{\tau}_{\mathrm{max}}=13600 psi,{\sigma}_{max}=30000 psi,{\delta}_{max}=0.25 in\end{array}} $$

Appendix 4 Pressure vessel design problem

$$ f(X)=0.6224{x}_1{x}_3{x}_4+1.7781{x}_2{x}_3^2+3.1661{x}_1^2{x}_4+19.84{x}_1^2{x}_3 $$
(Minimize)

subject to:

$$ {\displaystyle \begin{array}{c}{g}_1(X)=-{x}_1+0.0193{x}_3\le 0\\ {}{g}_2(X)=-{x}_2+0.0954{x}_3\le 0\\ {}{g}_3(X)=-\pi {x}_3^2-\left(\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)\pi {x}_3^3+1296000\le 0\\ {}\begin{array}{l}{g}_4(X)={x}_4-240\le 0\\ {}0\le {x}_i\le 100,i= 1, 2\\ {}10\le {x}_i\le 200,i= 3, 4\end{array}\end{array}} $$

Appendix 5 Speed reducer design problem

$$ {\displaystyle \begin{array}{l}f(X)=0.7854{x}_1{x}_2^2\left(3.3333{x}_3^2+14.9334{x}_3-43.0934\right)\\ {}-1.508{x}_1\left({x}_6^2+{x}_7^2\right)+7.4777\left({x}_6^3+{x}_7^3\right)+0.7854\left({x}_4{x}_6^2+{x}_5{x}_7^2\right)\end{array}} $$
(Minimize)

subject to:

$$ {\displaystyle \begin{array}{l}\begin{array}{c}{g}_1(X)=\frac{27}{x_1{x}_1^2{x}_3}-1\le 0\\ {}{g}_2(X)=\frac{397.5}{x_1{x}_2^2{x}_3^2}-1\le 0\\ {}{g}_3(X)=\frac{1.93{x}_4^3}{x_2{x}_3{x}_6^4}-1\le 0\\ {}{g}_4(X)=\frac{1.93{x}_5^3}{x_2{x}_3{x}_7^4}-1\le 0\\ {}{g}_5(X)=\frac{1.0}{110{x}_6^4}\sqrt{\left(\frac{745.0{x}_4}{x_2{x}_3}\right)+16.9\times {10}^6}-1\le 0\\ {}{g}_6(X)=\frac{1.0}{85{x}_7^4}\sqrt{\left(\frac{745.0{x}_5}{x_2{x}_3}\right)+157.5\times {10}^6}-1\le 0\\ {}{g}_7(X)=\frac{x_2{x}_3}{40}-1\le 0\\ {}{g}_8(X)=\frac{5{x}_2}{x_1}-1\le 0\\ {}{g}_9(X)=\frac{x_1}{12{x}_2}-1\le 0\\ {}{g}_{10}(X)=\frac{1.5{x}_6+1.9}{x_4}-1\le 0\\ {}{g}_{11}(X)=\frac{1.1{x}_7+1.9}{x_5}-1\le 0\end{array}\\ {} with\;2.6\le {x}_1\le 28,0.7\le {x}_2\le 0.8,\\ {}17\le {x}_3\le 28,7.3\le {x}_4\le 8.3,7.3\le {x}_5\le 8.3,\\ {}2.9\le {x}_63.9,5.0\le {x}_7\le 5.5,\end{array}} $$

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Liu, Z., Li, Z., Zhu, P. et al. A parallel boundary search particle swarm optimization algorithm for constrained optimization problems. Struct Multidisc Optim 58, 1505–1522 (2018). https://doi.org/10.1007/s00158-018-1978-3

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