Accelerated particle swarm optimization with explicit consideration of model constraints


Population based metaheuristic can benefit from explicit parallelization in order to address complex numerical optimization problems. Typical realistic problems usually involve non-linear functions and many constraints, making the identification of global optimal solutions mathematically challenging and computationally expensive. In this work, a GPU based parallelized version of the Particle Swarm Optimization technique is proposed. The main contribution is the explicit consideration of equality and inequality constraints of general type, rather than addressing only box constrained models as typically done in acceleration studies of optimization algorithms. The implementation is tested on a set of optimization problems that serve as benchmark. Speedups averaging 299x were obtained with a single GPU on a standard PC using the PyCUDA technology. Satisfactory feasibility and optimality rates are also achieved, although a standard parameterization was adopted for all the experiments. Additional results are reported on a small set of difficult problems involving bilinear non-linearities.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2


  1. 1.

    Adams II, T.A., Seider, W.D.: Practical optimization of complex chemical processes with tight constraints. Comput. Chem. Eng. 32(9), 2099–2112 (2008)

    Article  Google Scholar 

  2. 2.

    Adhya, N., Tawarmalani, M., Sahinidis, N.V.: A lagrangian approach to the pooling problem. Ind. Eng. Chem. Res. 38(5), 1956–1972 (1999)

    Article  Google Scholar 

  3. 3.

    Alba, E., Luque, G., Nesmachnow, S.: Parallel metaheuristics: recent advances and new trends. Int. Trans. Oper. Res. 20(1), 1–48 (2013)

    Article  Google Scholar 

  4. 4.

    Ali, M., Kaelo, P.: Improved particle swarm algorithms for global optimization. Appl. Math. Comput. 196(2), 578–593 (2008)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Bastos-Filho, C., Junior, M.O., Nascimento, D.: Running particle swarm optimization on graphic processing units. In: Search Algorithms and Applications. InTech (2011)

  6. 6.

    Blum, C., Puchinger, J., Raidl, G.R., Roli, A.: Hybrid metaheuristics in combinatorial optimization: a survey. Appl. Soft Comput. 11(6), 4135–4151 (2011)

    Article  Google Scholar 

  7. 7.

    Boussaïd, I., Lepagnot, J., Siarry, P.: A survey on optimization metaheuristics. Inf. Sci. 237, 82–117 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chen, T.Y., Chi, T.M.: On the improvements of the particle swarm optimization algorithm. Adv. Eng. Softw. 41(2), 229–239 (2010)

    Article  Google Scholar 

  9. 9.

    Chen, Y.W., Wang, L.C., Wang, A., Chen, T.L.: A particle swarm approach for optimizing a multi-stage closed loop supply chain for the solar cell industry. Robot. Comput. Integr. Manuf. 43, 111–123 (2017)

    Article  Google Scholar 

  10. 10.

    Coello Coello, C.A.: Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput. Methods Appl. Mech. Eng. 191(11–12), 1245–1287 (2002)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Crawford, B., Soto, R., Astorga, G., García, J., Castro, C., Paredes, F.: Putting continuous metaheuristics to work in binary search spaces. Complexity. (2017).

    MathSciNet  Article  Google Scholar 

  12. 12.

    Damiani, L., Diaz, A.I., Iparraguirre, J., Blanco, A.M.: Accelerated numerical optimization with explicit consideration of model constraints. In: Latin American High Performance Computing Conference, pp. 255–261. Springer, New York (2017)

    Google Scholar 

  13. 13.

    Eberhart, R., Kennedy, J.: A new optimizer using particle swarm theory. In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, 1995 (MHS’95), pp. 39–43. IEEE (1995)

  14. 14.

    Engelbrecht, A.P., Cleghorn, C.W.: Particle swarm optimization: a guide to effective, misconception free, real world use. In: Proceedings of the Genetic and Evolutionary Computation Conference Companion, pp. 831–857. ACM (2018)

  15. 15.

    Erbeyoğlu, G., Bilge, Ü.: Pso-based and sa-based metaheuristics for bilinear programming problems: an application to the pooling problem. J. Heuristics 22(2), 147–179 (2016)

    Article  Google Scholar 

  16. 16.

    Hung, Y., Wang, W.: Accelerating parallel particle swarm optimization via GPU. Optim. Methods Softw. 27(1), 33–51 (2012)

    Article  Google Scholar 

  17. 17.

    Kayhan, A.H., Ceylan, H., Ayvaz, M.T., Gurarslan, G.: Psolver: a new hybrid particle swarm optimization algorithm for solving continuous optimization problems. Expert Syst. Appl. 37(10), 6798–6808 (2010)

    Article  Google Scholar 

  18. 18.

    Laguna-Sánchez, G.A., Olguín-Carbajal, M., Cruz-Cortés, N., Barrón-Fernández, R., Álvarez-Cedillo, J.A.: Comparative study of parallel variants for a particle swarm optimization algorithm implemented on a multithreading GPU. J. Appl. Res. Technol. 7(3), 292–307 (2009)

    Article  Google Scholar 

  19. 19.

    Lang, J., Zhao, J.: Modeling and optimization for oil well production scheduling. Chin. J. Chem. Eng. 24(10), 1423–1430 (2016)

    Article  Google Scholar 

  20. 20.

    Liang, J., Runarsson, T.P., Mezura-Montes, E., Clerc, M., Suganthan, P.N., Coello, C.C., Deb, K.: Problem definitions and evaluation criteria for the cec 2006 special session on constrained real-parameter optimization. J. Appl. Mech. 41(8), 8–31 (2006)

    Google Scholar 

  21. 21.

    Marini, F., Walczak, B.: Particle swarm optimization (pso). a tutorial. Chemom. Intell. Lab. Syst. 149, 153–165 (2015)

    Article  Google Scholar 

  22. 22.

    Montain, M.E., Blanco, A.M., Bandoni, J.A.: Optimal drug infusion profiles using a particle swarm optimization algorithm. Comput. Chem. Eng. 82, 13–24 (2015)

    Article  Google Scholar 

  23. 23.

    Murtagh, B.A., Saunders, M.A.: Minos 5.4 user’s guide (preliminary). Tech. rep., Department of Operations Research, Stanford University (1987)

  24. 24.

    Mussi, L., Daolio, F., Cagnoni, S.: Evaluation of parallel particle swarm optimization algorithms within the \(\text{ CUDA }^{{\rm TM}}\) architecture. Inf. Sci. 181(20), 4642–4657 (2011)

    Article  Google Scholar 

  25. 25.

    Mussi, L., Nashed, Y.S., Cagnoni, S.: GPU-based asynchronous particle swarm optimization. In: Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation, pp. 1555–1562. ACM (2011)

  26. 26.

    NumPy: NumPy homepage. Accessed 05 Nov 2018

  27. 27.

    NVIDIA: PyCuda reference. Accessed 05 Nov 2018

  28. 28.

    Roberge, V., Tarbouchi, M.: Parallel particle swarm optimization on graphical processing unit for pose estimation. WSEAS Trans. Comput. 11(6), 170–179 (2012)

    Google Scholar 

  29. 29.

    Shokrian, M., High, K.A.: Application of a multi objective multi-leader particle swarm optimization algorithm on NLP and MINLP problems. Comput. Chem. Eng. 60, 57–75 (2014)

    Article  Google Scholar 

  30. 30.

    Souza, D., Teixeira, O., Monteiro, D., de Oliveira, R.: A cuda-based cooperative evolutionary multi-swarm optimization applied to engineering problems. In: Proc. of the XXXII Congress of the Brazilian Computing Society (2012)

  31. 31.

    Tan, Y., Ding, K.: A survey on GPU-based implementation of swarm intelligence algorithms. IEEE Trans. Cybern. 46(9), 2028–2041 (2016)

    Article  Google Scholar 

  32. 32.

    Tawarmalani, M., Sahinidis, N.V., Sahinidis, N.: Convexification and global optimization in continuous and mixed-integer nonlinear programming: theory, algorithms, software, and applications, vol. 65. Springer Science & Business Media, New York (2002)

    Google Scholar 

  33. 33.

    Wachowiak, M., Foster, A.L.: GPU-based asynchronous global optimization with particle swarm. J. Phys. Conf. Ser. 385, 012012 (2012)

    Article  Google Scholar 

  34. 34.

    Yiqing, L., Xigang, Y., Yongjian, L.: An improved PSO algorithm for solving non-convex NLP/MINLP problems with equality constraints. Comput. Chem. Eng. 31(3), 153–162 (2007)

    Article  Google Scholar 

  35. 35.

    Zhang, H., Rangaiah, G.P.: An efficient constraint handling method with integrated differential evolution for numerical and engineering optimization. Comput. Chem. Eng. 37, 74–88 (2012)

    Article  Google Scholar 

  36. 36.

    Zhou, Y., Tan, Y.: GPU-based parallel particle swarm optimization. In: IEEE Congress on Evolutionary Computation, 2009 (CEC’09), pp. 1493–1500. IEEE (2009)

Download references


This research was partially supported by grants from Consejo Nacional de InvestigacionesCientíficas y Técnicas (CONICET) and Universidad Tecnológica Nacional (UTN) of Argentina. The authors also gratefully acknowledge the support of NVIDIA Corporation with the donation of the TITAN X GPU used in this research.

Author information



Corresponding author

Correspondence to Javier Iparraguirre.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.



Model U1: Branin RCOS function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) =a\left( x_{2}-bx_1^2+cx_{1}-d\right) ^{2}+e\left( 1-f\right) \left( \cos \left( x_{1}\right) \right) +e\\&Where\ a=1, b=5.1/\left( 4\pi ^{2}\right) , c=5/\pi , d=6, f=1/\left( 8\pi \right) \\&Bounds\ -5\le x_{1}\le 10,\ 0\le x_{1}\le 15\\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-0.397887\\&\overrightarrow{x}^{*}=\left( 0, 0\right) \end{aligned}$$

Model U2: Shubert function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) \\&\quad =\left( \sum _{j=1}^5j\cos \left( j+1\right) x_{1}+j\right) \left( \sum _{j=1}^5j\cos \left( j+1\right) x_{1}+j\right) \\&Bounds\ -10\le x_{j}\le 10 \left( j=1,2\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-186\\&\overrightarrow{x}^{*}=\left( 0,0\right) \end{aligned}$$

Model U3: Michalewicz function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) \\&\quad =-\sum _{j=1}^2\sin \left( x_{j}\right) \left( \sin \left( jx_j^2/\pi \right) \right) ^{2m};\ m=10\\&Bounds\ 0\le x_{j}\le \pi \left( j=1,2\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-1.8013\\&\overrightarrow{x}^{*}=\left( 2.20, 1.57\right) \end{aligned}$$

Model U4: Colville function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) =100\left( x_1^2-x_{2}\right) ^{2} \\&\quad +\left( x_{1}-1\right) ^{2}+\left( x_{3}-1\right) ^{2}+90\left( x_3^2-x_{4}\right) ^{2}\\&\quad +10.1\left[ \left( x_{2}-1\right) ^{2}+\left( x_{4}-1\right) ^{2}\right] +19.8\left( x_{2}-1\right) \left( x_{4}-1\right) \\&Bounds\ -10\le x_{j}\le 10 \left( j=1,\ldots ,4\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0\\&\overrightarrow{x}^{*}=\left( 1,1,1,1\right) \end{aligned}$$

Model U5: Spherical function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) =-\sum _{j=1}^Dx_j^2\ ;\ D=10\\&Bounds\ -100\le x_{j}\le 100 \left( j=1,\ldots ,10\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0\\&\overrightarrow{x}^{*}=\left( 0,0,0,0,0,0,0,0,0,0\right) \end{aligned}$$

Model U6: Quadric function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) =\sum _{i=1}^D\left( \sum _{j=1}^D x_j^2\right) ^2; \ D=10\\&Bounds\ -100\le x_{j}\le 100 \left( j=1,\ldots ,10\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0\\&\overrightarrow{x}^{*}=\left( 0,0,0,0,0,0,0,0,0,0\right) \end{aligned}$$

Model U7: Rosenbrock function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) \\&\quad =\sum _{j=1}^{D-1}\left( 100\left( x_j^2-x_{j+1}\right) ^{2}+\left( x_{j}-1\right) ^{2}\right) ; \ D=10\\&Bounds\ -5\le x_{j}\le 10 \left( j=1,\ldots ,10\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0\\&\overrightarrow{x}^{*}=\left( 1,1,1,1,1,1,1,1,1,1\right) \end{aligned}$$

Model U8: Griewank function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) \\&\quad =1+\frac{1}{4000}\sum _{j=1}^{D}\left( x_j^2\right) -\prod _{j=1}^D\cos \left( \frac{x_{j}}{\sqrt{j}}\right) ; \ D=10\\&Bounds\ -600\le x_{j}\le 600 \left( j=1,\ldots ,10\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0\\&\overrightarrow{x}^{*}=\left( 0,0,0,0,0,0,0,0,0,0\right) \end{aligned}$$

Model U9: Rastrigin function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) =10D+\sum _{j=1}^D\left[ x_j^2-10\cos \left( 2\pi x_{j}\right) \right] \\&Bounds\ -5.12\le x_{j}\le 5.12 \left( j=1,\ldots ,10\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0\\&\overrightarrow{x}^{*}=\left( 0,0,0,0,0,0,0,0,0,0\right) \end{aligned}$$

Model U10: Bukin function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) =100\sqrt{\mid x_{2}-0.01x_1^2\mid }+0.01\mid x_{1}+10\mid \\&Bounds\ -15\le x_{1}\le 5,\ -3\le x_{2}\le 3\\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0\\&\overrightarrow{x}^{*}=\left( -10, 1\right) \end{aligned}$$

Model U11: Schwefel function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) =\sum _{j=1}^D\left( x_{j}+0.5\right) ^{2}\\&Bounds\ -10\le x_{j}\le 10 \left( j=1,\ldots ,10\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0\\&\overrightarrow{x}^{*}=\left( -0.5,-0.5,-0.5,-0.5,-0.5,-0.5,-0.5,-0.5,-0.5,-0.5\right) \end{aligned}$$

Model U12: Step function

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) =\sum _{j=1}^D-x_{j}\sin \left( \sqrt{\mid x_{j}\mid }\right) \\&Bounds\ -500\le x_{j}\le 500 \left( j=1,\ldots ,10\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-4189.829\\&\overrightarrow{x}^{*}=(420.968, 420.968, 420.968, 420.968, 420.968,\\&420.968, 420.968, 420.968, 420.968, 420.968)\\ \end{aligned}$$

Model C1

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) =5\sum _{j=1}^4x_{j}-5\sum _{j=1}^4\ x_{j}^{2}-\sum _{j=5}^{13}x_{j} \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = 2 x_{1}+2 x_{2}+x_{10}+x_{11}-10 \le 0 \\&g_{2}\left( \overrightarrow{x}\right) = 2 x_{1}+2 x_{3}+x_{10}+x_{12}-10 \le 0 \\&g_{3}\left( \overrightarrow{x}\right) = 2 x_{2}+2 x_{3}+x_{11}+x_{12}-10 \le 0 \\&g_{4}\left( \overrightarrow{x}\right) = -8 x_{1}+x_{10}\le 0 \\&g_{5}\left( \overrightarrow{x}\right) = -8 x_{2}+x_{11}\le 0 \\&g_{6}\left( \overrightarrow{x}\right) = -8 x_{3}+x_{12}\le 0 \\&g_{7}\left( \overrightarrow{x}\right) = -2 x_{4}-x_{5}+x_{10}\le 0 \\&g_{8}\left( \overrightarrow{x}\right) = -2 x_{6}-x_{7}+x_{11}\le 0 \\&g_{9}\left( \overrightarrow{x}\right) = -2 x_{8}-x_{9}+x_{12}\le 0 \\&Bounds\ 0\le x_{j}\le 1 \left( j=1,\ldots ,9\right) , 0\le x_{j}\le 100 \left( j=10,11,12\right) \ and 0 \le x_{13} \le 1 \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-15\\&\overrightarrow{x}^{*}=\left( 1,1,1,1,1,1,1,1,1,3,3,3,1\right) \end{aligned}$$

Model C2

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = -\left( \sqrt{D}^{D}\right) \prod _{j=1}^{D}x_{j}\\ \end{aligned}$$

Subject to:

$$\begin{aligned}&h_{1}\left( \overrightarrow{x}\right) = \sum _{j=1}^D x_j^2-1=0 \\&Bounds\ 0\le x_{j}\le 1 \left( j=1,\ldots ,10\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-1.00050010001000\\&\overrightarrow{x}^{*}=(0.31624357647283069, 0.31624357647283069, 0.31624357647283069,\\&0.31624357647283069, 0.31624357647283069, 0.31624357647283069,\\&0.31624357647283069, 0.31624357647283069, 0.31624357647283069,\\&0.31624357647283069) \end{aligned}$$

Model C3

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = 5.3578547x_3^2+0.000001x_1^3+2x_{2}+\left( 0.000002/3\right) x_3^2 \\ \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = 85.334407+0.0056858x_{2}x_{5}\\&\quad +0.0006262x_{1}x_{4}-0.0022053 x_{3}x_{5}-92 \le 0 \\&g_{2}\left( \overrightarrow{x}\right) = -85.334407-0.0056858x_{2}x_{5}\\&\quad -0.0006262x_{1}x_{4}+0.0022053 x_{3}x_{5} \le 0 \\&g_{3}\left( \overrightarrow{x}\right) = 80.51249+0.0071317x_{2}x_{5}\\&\quad +0.0029955x_{1}x_{2}-0.0021813x_3^2-110 \le 0 \\&g_{4}\left( \overrightarrow{x}\right) = -80.51249-0.0071317x_{2}x_{5}\\&\quad -0.0029955x_{1}x_{2}-0.0021813x_3^2+90 \le 0 \\&g_{5}\left( \overrightarrow{x}\right) = 9.300961+0.0047026x_{3}x_{5}\\&\quad +0.0012547x_{1}x_{3}+0.0019085 x_{3}x_{4}-25 \le 0 \\&g_{6}\left( \overrightarrow{x}\right) =-9.300961-0.0047026x_{3}x_{5}\\&\quad -0.0012547x_{1}x_{3}-0.0019085 x_{3}x_{4}+20 \le 0 \\&Bounds\ 78\le x_{1}\le 102,\ 33\le x_{2}\le 45,\ and 27\le x_{j}\le 45,\ \left( j=3,4,5\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-30665.53867178332\\&\overrightarrow{x}^{*}=\left( 78, 33, 29.9952560256815985, 45, 36.7758129057882073 \right) \end{aligned}$$

Model C4

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = 3x_{1}+0.000001x_1^3+2x_{2}+\left( 0.000002/3\right) x_2^3\\ \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = -x_{4}+x_{3}-0.55 \le 0 \\&g_{2}\left( \overrightarrow{x}\right) = -x_{3}+x_{4}-0.55 \le 0 \\&h_{1}\left( \overrightarrow{x}\right) = 1000\sin \left( -x_{3}-0.25\right) \\&\quad +1000\sin \left( -x_{4}-0.25\right) +894.8-x_{1} =0 \\&h_{2}\left( \overrightarrow{x}\right) = 1000\sin \left( x_{3}-0.25\right) \\&\quad +1000\sin \left( x_{3}-x_{4}-0.25\right) +894.8-x_{2} =0 \\&h_{3}\left( \overrightarrow{x}\right) = 1000\sin \left( x_{4}-0.25\right) \\&\quad +1000\sin \left( x_{4}-x_{3}-0.25\right) +1294.8=0 \\&Bounds\ 0\le x_{1}\le 1200,\ 0\le x_{2}\le 1200,\ -0.55\le x_{3}\le 0.55,\ and -0.55\le x_{4}\le 0.55\\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =5126.4967140071\\&\overrightarrow{x}^{*}=(679.945148297028709, 1026.06697600004691,\\&0.118876369094410433, -0.39623348521517826) \end{aligned}$$

Model C5

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = \left( x_{1}-10\right) ^{3}+ \left( x_{2}-20\right) ^{3}\\ \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = -\left( x_{1}-5\right) ^{2}-\left( x_{2}-5\right) ^{2} +100\le 0 \\&g_{2}\left( \overrightarrow{x}\right) = \left( x_{1}-6\right) ^{2}+\left( x_{2}-5\right) ^{2} -82.81\le 0 \\&Bounds\ 13\le x_{1}\le 100 and 0\le x_{2}\le 100 \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-6961.81387558015\\&\overrightarrow{x}^{*}=\left( 14.09500000000000064, 0.8429607892154795668\right) \end{aligned}$$

Model C6

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = x_1^2+x_2^2+x_{1}x_{2}-14x_{1}-16x_{2}\\&\quad +\left( x_{3}-10\right) ^{2}+4\left( x_{4}-5\right) ^{2}+\left( x_{5}-3\right) ^{2}\\&\quad +2\left( x_{6}-1\right) ^{2}+5x_7^2+7\left( x_{8}-11\right) ^{2}+2\left( x_{9}-10\right) ^{2}+\left( x_{10}-7\right) ^{2}+45 \\ \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = -105+4x_{1}+5x_{2}-3x_{7}+9x_{8} \le 0 \\&g_{2}\left( \overrightarrow{x}\right) = 10x_{1}-8x_{2}-17x_{7}+2x_{8} \le 0 \\&g_{3}\left( \overrightarrow{x}\right) = -8x_{1}+2x_{2}+5x_{9}-2x_{10}-12 \le 0 \\&g_{4}\left( \overrightarrow{x}\right) = 3\left( x_{1}-2\right) ^{2}+4\left( x_{2}-3\right) ^{2}+2x_3^2-7x_{4}-120 \le 0 \\&g_{5}\left( \overrightarrow{x}\right) = 5x_1^2+8x_{2}+\left( x_{3}-6\right) ^{2}-2x_{4}-40 \le 0 \\&g_{6}\left( \overrightarrow{x}\right) = x_1^2+2\left( x_{2}-2\right) ^{2}-2x_{1}x_{2}+14x_{5}-6x_{6} \le 0 \\&g_{7}\left( \overrightarrow{x}\right) = 0.5\left( x_{1}-8\right) ^{2}+2\left( x_{2}-4\right) ^{2}+3x_5^2-x_{6}-30 \le 0 \\&g_{8}\left( \overrightarrow{x}\right) = -3x_{1}+6x_{2}+12\left( x_{9}-8\right) ^{2}-7x_{10} \le 0 \\&Bounds\ -10\le x_{j}\le 10 \left( j=1,\ldots ,10\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =24.30620906818\\&\overrightarrow{x}^{*}=(2.17199634142692, 2.3636830416034, 8.77392573913157, 5.09598443745173,\\&0.990654756560493, 1.43057392853463, 1.32164415364306, 9.82872576524495,\\&8.2800915887356, 8.375926647734)\\ \end{aligned}$$

Model C7

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = -\frac{\sin ^{3}\left( 2\pi x_{1}\right) \sin \left( 2\pi x_{2}\right) }{x_1^3 \left( x_{1}+x_{2}\right) } \\ \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = x_1^2 -x_{2}+1 \le 0 \\&g_{2}\left( \overrightarrow{x}\right) = 1-x_{1}+\left( x_{2}-4\right) ^{2} \le 0 \\&Bounds\ 0\le x_{j}\le 1 \left( j=1,2\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) = -0.0958250414180359 \\&\overrightarrow{x}^{*}=\left( 1.22797135260752599, 4.24537336612274885\right) \end{aligned}$$

Model C8

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) \\&\quad =\left( x_{1}-10\right) ^{2}+5\left( x_{2}-12\right) ^{2}+x_3^4+3\left( x_{4}-11\right) ^{2}\\&\quad +10x_5^6+7x_6^2+x_7^4-4x_{6}x_{7}-10x_{6}-8x_{7} \\ \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = -127+2x_1^2+3x_2^4+x_{3}+4x_4^2+5x_{5} \le 0 \\&g_{2}\left( \overrightarrow{x}\right) = -282+7x_{1}+3x_{2}+10x_3^2+x_{4}-x_{5} \le 0 \\&g_{3}\left( \overrightarrow{x}\right) = -196+23x_{1}+x_2^2+6x_6^2-8x_{7} \le 0 \\&g_{4}\left( \overrightarrow{x}\right) = 4x_1^1+x_2^2-3x_{1}x_{2}+2x_3^2+5x_{6}-11x_{7} \le 0 \\&Bounds\ -10\le x_{j}\le 10 \left( j=1,\ldots ,7 \right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =680.630057374402\\&\overrightarrow{x}^{*}=(2.33049935147405174, 1.9137236847114592, -0.477541399510615805,\\&4.36572624923625874,-0.624486959100388983, 1.03813099410962173, 1.5942266780671519)\\ \end{aligned}$$

Model C9

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = x_{1}+x_{2}+x_{3}\\ \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = -1+0.0025\left( x_{4}+x_{6}\right) \le 0 \\&g_{2}\left( \overrightarrow{x}\right) = -1+0.0025\left( x_{5}+x_{7}-x_{4}\right) \le 0 \\&g_{3}\left( \overrightarrow{x}\right) = -1+0.01\left( x_{8}-x_{5}\right) \le 0 \\&g_{4}\left( \overrightarrow{x}\right) = -x_{1}x_{6}+833.33252x_{4}+100x_{1}-83333.333\le 0 \\&g_{5}\left( \overrightarrow{x}\right) = -x_{2}x_{7}+1250x_{5}+x_{2}x_{4}-1250x_{4} \le 0 \\&g_{6}\left( \overrightarrow{x}\right) = -x_{3}x_{8}+1250000+x_{3}x_{5}-2500x_{5} \le 0 \\&Bounds\ 100\le x_{1}\le 1000, \ 1000\le x_{j}\le 10000 \left( j=2,3\right) , \ 10\le x_{j}\\&\quad \le 1000 \left( j=4,\ldots ,8\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =7049.24802052867\\&\overrightarrow{x}^{*}=(579.306685017979589, 1359.97067807935605, 5109.97065743133317, \\&182.01769963061534, 295.601173702746792, 217.982300369384632, 286.41652592786552, \\&395.601173702746735)\\ \end{aligned}$$

Model C10

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = x_1^2+\left( x_{2}-1\right) ^{2} \\ \end{aligned}$$

Subject to:

$$\begin{aligned}&h_{1}\left( \overrightarrow{x}\right) =x_{2}-x_1^2 =0 \\&Bounds\ -1\le x_{1}\le 1, \ -1\le x_{2}\le 1 \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0.7499 \\&\overrightarrow{x}^{*}=\left( -0.707036070037170616, 0.500000004333606807 \right) \end{aligned}$$

Model C11

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = e^{x_{1}x_{2}x_{3}x_{4}x_{5}} \\ \end{aligned}$$

Subject to:

$$\begin{aligned}&h_{1}\left( \overrightarrow{x}\right) = x_1^2+x_2^2+x_3^2+x_4^2+x_5^2-10 =0 \\&h_{2}\left( \overrightarrow{x}\right) = x_{2}x_{3}-5x_{4}x_{5} =0 \\&h_{3}\left( \overrightarrow{x}\right) = x_1^3+x_2^3+1 =0 \\&Bounds\ -2.3\le x_{j}\le 2.3, \left( j=1,2\right) \ -3.2\le x_{j}\le 3.2 \left( j=3,4,5 \right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =0.053941514041898 \\&\overrightarrow{x}^{*}=\left( 1.71714224003, 1.59572124049468, -0.763659881912867, -0.76365986736498 \right) \end{aligned}$$

Model C12

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = \sum _{j=1}^{10}x_{j}\left( c_{j}+\ln {\frac{x_{j}}{\sum _{j=1}^{10}x_{j}}}\right) \\ \end{aligned}$$

Subject to:

$$\begin{aligned}&h_{1}\left( \overrightarrow{x}\right) = x_{1}+2x_{2}+2x_{3}+x_{6}+x_{10}-2 =0 \\&h_{2}\left( \overrightarrow{x}\right) = x_{4}+2x_{5}+x_{6}+x_{7}-1 =0 \\&h_{3}\left( \overrightarrow{x}\right) = x_{3}+x_{7}+x_{8}+2x_{9}+x_{10}-1 =0 \\&Bounds\ 0\le x_{j}\le 10 \left( j=1,\ldots ,10\right) \\&Where\ c_{1}=-6.089, c_{2}=-17.164, c_{3}=-34.054, c_{4}=-5.914, c_{5}=-24.721, \\&c_{6}=-14.986, c_{7}=-24.1, c_{8}=-10.708, c_{9}=-26.662, c_{10}=-22.179 \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-47.7648884594915\\&\overrightarrow{x}^{*}= \ (0.0406684113216282, 0.147721240492452, 0.783205732104114, \\&\ \ \ \ \ \ \ \ \ 0.00141433931889084, 0.485293636780388, 0.000693183051556082, \\&\ \ \ \ \ \ \ \ \ 0.0274052040687766, 0.0179509660214818, 0.0373268186859717, \\&\ \ \ \ \ \ \ \ \ 0.0968844604336845) \end{aligned}$$

Model C13

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = 1000-x_1^2-2x_2^2-x_3^2-x_{1}x_{2}-x_{1}x_{3}\\ \end{aligned}$$

Subject to:

$$\begin{aligned}&h_{1}\left( \overrightarrow{x}\right) = x_1^2+x_2^2+x_3^2-25 =0 \\&h_{2}\left( \overrightarrow{x}\right) = 8x_{1}+14x_{2}+7x_{3}-56 =0 \\&Bounds\ 0\le x_{j}\le 10 \left( j=1,2,3\right) \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =961.715022289961\\&\overrightarrow{x}^{*}=\left( 3.51212812611795133, 0.216987510429556135 \right) \end{aligned}$$

Model C14

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = -0.5\left( x_{1}x_{4}-x_{2}x_{3}+x_{3}x_{9}-x_{5}x_{9}+x_{5}x_{8}-x_{6}x_{7}\right) \\ \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = x_3^2+x_4^2-1x_3^2+x_4^2-1 \le 0 \\&g_{2}\left( \overrightarrow{x}\right) = x_9^2-1 \le 0 \\&g_{3}\left( \overrightarrow{x}\right) = x_5^2+x_6^2-1 \le 0 \\&g_{4}\left( \overrightarrow{x}\right) = x_1^2+\left( x_2-x_9\right) ^{2}-1 \le 0 \\&g_{5}\left( \overrightarrow{x}\right) = \left( x_1-x_5\right) ^{2}+\left( x_2-x_6\right) ^{2}-1 \le 0 \\&g_{6}\left( \overrightarrow{x}\right) = \left( x_1-x_7\right) ^{2}+\left( x_2-x_8\right) ^{2}-1 \le 0 \\&g_{7}\left( \overrightarrow{x}\right) = \left( x_3-x_5\right) ^{2}+\left( x_4-x_6\right) ^{2}-1 \le 0 \\&g_{8}\left( \overrightarrow{x}\right) = \left( x_3-x_7\right) ^{2}+\left( x_4-x_8\right) ^{2}-1 \le 0 \\&g_{9}\left( \overrightarrow{x}\right) = x_7^2\left( x_4-x_8\right) ^{2}-1 \le 0 \\&g_{10}\left( \overrightarrow{x}\right) = x_2x_3-x_1x_4 \le 0 \\&g_{11}\left( \overrightarrow{x}\right) = -x_3x_9 \le 0 \\&g_{12}\left( \overrightarrow{x}\right) = x_5x_9 \le 0 \\&g_{13}\left( \overrightarrow{x}\right) = x_6x_7-x_5x_8 \le 0 \\&Bounds\ -10\le x_{j}\le 10 \left( j=1,\ldots ,8\right) 0\le x_{9}\le 20 \\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-0.866025403784439\\&\overrightarrow{x}^{*}=(-0.657776192427943196, -0.15341877348243854, 0.323413871675240938, \\&-0.946257611651304398, -0.657776194376798906, -0.753213434632691414, \\&0.323413874123576972,-0.346462947962331735, 0.59979466285217542) \end{aligned}$$

Model C15

$$\begin{aligned}&Minimize\ f\left( \overrightarrow{x}\right) = x_1-x_2\\ \end{aligned}$$

Subject to:

$$\begin{aligned}&g_{1}\left( \overrightarrow{x}\right) = -2x_1^4+8x_1^3-8x_1^2+x_2-2 \le 0 \\&g_{2}\left( \overrightarrow{x}\right) = -4x_1^4+32x_1^3-88x_1^2+96x_1+x_2-36\le 0 \\&Bounds\ 0\le x_{1}\le 3, \ 0\le x_{2}\le 4\\&Global\ minimum\ f\left( \overrightarrow{x}^{*}\right) =-5.50801327159536\\&\overrightarrow{x}^{*}=\left( 2.32952019747762, 3.17849307411774 \right) \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Damiani, L., Diaz, A.I., Iparraguirre, J. et al. Accelerated particle swarm optimization with explicit consideration of model constraints. Cluster Comput 23, 149–164 (2020).

Download citation


  • Numerical optimization
  • Particle swarm optimization
  • GPU