Accelerated particle swarm optimization with explicit consideration of model constraints

Abstract

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.

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Acknowledgements

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.

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Correspondence to Javier Iparraguirre.

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Appendix

Appendix

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:

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

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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). https://doi.org/10.1007/s10586-019-02933-1

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Keywords

  • Numerical optimization
  • Particle swarm optimization
  • GPU