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Danger theory based artificial immune system solving dynamic constrained single-objective optimization

  • Methodologies and Application
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Abstract

In this paper, we propose an artificial immune system (AIS) based on the danger theory in immunology for solving dynamic nonlinear constrained single-objective optimization problems with time-dependent design spaces. Such proposed AIS executes orderly three modules—danger detection, immune evolution and memory update. The first module identifies whether there are changes in the optimization environment and decides the environmental level, which helps for creating the initial population in the environment and promoting the process of solution search. The second module runs a loop of optimization, in which three sub-populations each with a dynamic size seek simultaneously the location of the optimal solution along different directions through co-evolution. The last module stores and updates the memory cells which help the first module decide the environmental level. This optimization system is an on-line and adaptive one with the characteristics of simplicity, modularization and co-evolution. The numerical experiments and the results acquired by the nonparametric statistic procedures, based on 22 benchmark problems and an engineering problem, show that the proposed approach performs globally well over the compared algorithms and is of potential use for many kinds of dynamic optimization problems.

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Acknowledgments

This work is supported by National Natural Science Foundation NSFC(61065010) and Doctoral Fund of Ministry of Education of China (20125201110003). The first and the second authors are partially supported by EU FP7 HAZCEPT (318907) projects.

Author information

Authors and Affiliations

  1. Institute of System Science and Information Technology, College of Science, Guizhou University, Guiyang, Guizhou, China

    Zhuhong Zhang & Min Liao

  2. School of Computer Science, University of Lincoln, Lincoln, UK

    Shigang Yue

  3. Institute of Intelligent Information Processing, College of Computer Science and Information, Guizhou University, Guiyang, Guizhou, China

    Fei Long

Authors
  1. Zhuhong Zhang
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  2. Shigang Yue
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  3. Min Liao
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  4. Fei Long
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Corresponding author

Correspondence to Zhuhong Zhang.

Additional information

Communicated by F. Herrera.

Appendix

Appendix

Test problems (k = 0.5)

$$\begin{aligned} &G01.\,Min\,f(X) = 5\sum\limits^{4}_{i=1}X_{i}-5\sum\limits^{4}_{i=1}X_{i}^{2}-\sum\limits^{13}_{i=5}X_{i}\\ &s.t.,\\ &\quad g_{1}(Y) = 2Y_{1} + 2Y_{2} + Y_{10} + Y_{11}-10\leq 0, \\ &\quad g_{2}(Y) = 2Y_{1} + 2Y_{3} + Y_{10} + Y_{12}-10\leq 0,\\ &\quad g_{3}(Y) = 2Y_{2} + 2Y_{3} + Y_{11} + Y_{12}-10\leq 0,\\ &\quad g_{4}(Y) =-8Y_{1}+ Y_{10}\leq 0,\\ &\quad g_{5}(Y) = -8Y_{2} + Y_{11}\leq 0,\\ &\quad g_{6}(Y) = -8Y_{3} + Y_{12}\leq 0,\\ &\quad g_{7}(Y) = -2Y_{4} -Y_{5} + Y_{10}\leq 0,\\ &\quad g_{8}(Y)= -2Y_{6} -Y_{7} + Y_{11}\leq 0, \\&\quad g_{9}(Y) = -2Y_{8}-Y_{9} + Y_{12}\leq 0,\\ &\quad X_{i}=cos\frac{k\pi t}{2}x_{i},\quad Y_{i}=x_{i},\quad -sin \frac{k\pi t}{2} \leq x_{i}\leq 1+sin \frac{k\pi t}{2},\quad1\leq i\leq 9, \\ &\quad -sin \frac{k\pi t}{2}\leq x_{i}\leq 100+sin \frac{k\pi t}{2},\quad 10\leq i\leq 12, -sin \frac{k\pi t}{2}\leq x_{13}\leq 1+sin \frac{k\pi t}{2}.\\ \end{aligned} $$
$$ \begin{aligned} &G02.\,Min\,f(X)=-\left|\frac{\sum_{i=1}^{p(t)}cos^{4}(X_{i})-2\prod_{i=1}^{p(t)}cos^{2}(X_{i})} {\sqrt{\sum_{i=1}^{p(t)}iX_{i}^{2}}}\right|\\ &s.t.,\\ &\quad g_{1}(Y) = 0.75-\prod_{i=1}^{p(t)}Y_{i}\leq 0, \\ &\quad g_{2}(Y) = \sum\limits_{i=1}^{p(t)}Y_{i}-7.5p(t)\leq 0,\\&\quad X_{i}=x_{i},\quad Y_{i}=\left(cos\frac{k\pi t}{100}\right)x_{i},\quad -sin \frac{k\pi t}{2} < x_{i}\leq 10+sin \frac{k\pi t}{2}, \\ &\quad 1\leq i \leq p(t),\quad p(t)= 20+\bigg\lfloor20\left|sin\frac{k\pi t}{2}\right|\bigg\rfloor.\\ \end{aligned}$$
$$ \begin{aligned}&G03.\,Min\,f(X) =-p(t)^{p(t)/2}\prod\limits_{i=1}^{p(t)}X_{i}\\ &s.t.,\\ &\quad h_{1}(Y) =\sum\limits_{i=1}^{p(t)}Y_{i}^{2}-1 = 0,\\ &\quad X_{i}=x_{i}+sin \frac{\pi t}{100p(t)},\quad Y_{i}=\left(cos\frac{k\pi t}{100p(t)}\right)x_{i},\\ &\quad -sin \frac{k\pi t}{2}\leq x_{i}\leq 1+sin \frac{k\pi t}{2},\quad 1\leq i \leq p(t), p(t)= 20+\bigg\lfloor20\left|sin\frac{k\pi t}{2}\right|\bigg\rfloor.\\ \end{aligned} $$
$$ \begin{aligned}&G04.\,Min\,f(X) = 5.3578547X_{3}^{2}+ 0.8356891X_{1}X_{5} + 37.293239X_{1}-40792.141\\ &s.t., \\ & \quad g_{1}(Y) = 85.334407 + 0.0056858Y_{2}Y_{5} + 0.0006262Y_{1}Y_{4} -0.0022053Y_{3}Y_{5}-92\leq 0,\\ &\quad g_{2}(Y) =-5.334407-0.0056858Y_{2}Y_{5}-0.0006262Y_{1}Y_{4} + 0.0022053Y_{3}Y_{5}\leq 0,\\ &\quad g_{3}(Y) = 80.51249 + 0.0071317Y_{2}Y_{5} + 0.0029955Y_{1}Y_{2} + 0.0021813Y_{3}^{2}-110\leq 0,\\ &\quad g_{4}(Y) = -0.51249-0.0071317Y_{2}Y_{5}-0.0029955Y_{1}Y_{2}-0.0021813Y_{3}^{2}+90\leq 0,\\ &\quad g_{5}(Y) = 9.300961 + 0.0047026Y_{3}Y_{5} + 0.0012547Y_{1}Y_{3} + 0.0019085Y_{3}Y_{4}-25\leq 0,\\ &\quad g_{6}(Y) =-9.300961-0.0047026Y_{3}Y_{5}-0.0012547Y_{1}Y_{3}-0.0019085Y_{3}Y_{4} + 20\leq 0,\\ &\quad X_{i}=x_{i},\quad Y_{i}=x_{i}+\frac{t(b_{i}(t)-a_{i}(t))}{5000},\quad 78-sin \frac{k\pi t}{2}\leq x_{1}\leq 102+sin \frac{k\pi t}{2}, \\ &\quad 33-sin \frac{k\pi t}{2}\leq x_{2}\leq 45+sin \frac{k\pi t}{2},\quad 27-sin \frac{k\pi t}{2}\leq x_{i}\leq 45+sin \frac{k\pi t}{2},\quad 3\leq i \leq 5.\\ \end{aligned} $$
$$ \begin{aligned}&G05.\,Min\,f(X) = 3X_{1} + 0.000001X_{1}^{3}+ 2X_{2} + (0.000002/3)X_{2}^{3}\\& s.t., \\&\quad g_{1}(Y) =-Y_{4} + Y_{3}-0.55\leq 0, \quad g_{2}(Y) = -Y_{3} + Y_{4}-0.55\leq 0,\\ &\quad h_{1}(Y) = 1000 sin(-Y_{3}-0.25) + 1000 sin(-Y_{4}-0.25) + 894.8-Y1 = 0,\\ &\quad h_{2}(Y) = 1000 sin(Y_{3}-0.25) + 1000 sin(Y_{3}-Y_{4}-0.25) + 894.8-Y_{2} = 0,\\ &\quad h_{3}(Y) = 1000 sin(Y_{4} -0.25) + 1000 sin(Y_{4}-Y_{3}-0.25) + 1294.8 = 0, \\ &\quad X_{i}=\left(1+\frac{k\pi t}{100p(t)}\right)x_{i},\quad Y_{i}=x_{i}+sin\left(\frac{k\pi t}{1000p(t)}\right),\quad -sin \frac{k\pi t}{2}\leq x_{1}\leq 1200+sin \frac{k\pi t}{2},\\ &\quad -sin \frac{k\pi t}{2}\leq x_{2}\leq 1200+sin \frac{k\pi t}{2}, -0.55-sin \frac{k\pi t}{2}\leq x_{3}\leq 0.55+sin \frac{k\pi t}{2}, \\ &\quad -0.55-sin \frac{k\pi t}{2}\leq x_{4}\leq 0.55+sin \frac{k\pi t}{2}.\\ \end{aligned} $$
$$ \begin{aligned}&G06.\,Min\,f(X) = (X_{1}-10)^{3}+(X_{2} -20)^{3}\\ &s.t.,\\ &\quad g_{1}(Y) = -(Y_{1}-5)^{2}-(Y_{2}-5)^{2}+ 100\leq 0,\\ &\quad g_{2}(Y) = (Y_{1}-6)^{2} + (Y_{2}-5)^{2}-82.81\leq 0,\\ &\quad X_{i}=x_{i}+sin \frac{\pi t}{2},\quad Y_{i}=x_{i}+\frac{t(b_{i}(t)-a_{i}(t))}{5000},\quad 13-sin \frac{k\pi t}{2}\leq x_{1}\leq 100+sin \frac{k\pi t}{2},\\ &\quad -sin \frac{k\pi t}{2}\leq x_{2}\leq 100+sin \frac{k\pi t}{2}.\\ \end{aligned} $$
$$ \begin{aligned}&G07.\,Min\,f(X) = X_{1}^{2}+ X_{2}^{ 2} + X_{1}X_{2}-14X_{1}-16X_{2} + (X_{3}-10)^{2} + 4(X_{4}-5)^{2}\\ &\,\,\,\,\,\,\,\,\,\,\,\,\,+ (X_{5}-3)^{2}+2(X_{6}-1)^{2} + 5X^{2}_{7} + 7(X_{8}-11)^{2} + 2(X_{9}-10)^{2} + (X_{10}-7)^{2} + 45\\ &s.t.,\\ &\quad g _{1}(Y) =-105 + 4Y_{1} + 5Y_{2}-3Y_{7} + 9Y_{8}\leq 0,\\ &\quad g_{2}(Y) = 10Y_{1}-8Y_{2}-17Y_{7} + 2Y_{8}\leq 0,\\ &\quad g_{3}(Y) =-8Y_{1} + 2Y_{2} + 5Y_{9}-2Y_{10}-12 \leq 0,\\ &\quad g_{4}(Y) = 3(Y_{1} -2)^{2} + 4(Y_{2}-3)^{2} + 2Y^{2}_{ 3}-7Y_{4} -120\leq 0,\\ &\quad g_{5}(Y) = 5Y^{2}_{1} + 8x_{2} + (Y_{3}-6)^{2}-2Y_{4} -40\leq 0,\\ &\quad g_{6}(Y) = Y^{2}_{1} + 2(Y_{2}-2)^{2}-2Y_{1}Y_{2} + 14Y_{5}-6Y_{6}\leq 0,\\ &\quad g_{7}(Y) = 0.5(Y_{1}-8)^{2} + 2(Y_{2}-4)^{2} + 3Y^{2}_{5}-Y_{6}-30 \leq 0,\\ &\quad g_{8}(Y) = -3Y_{1} + 6Y_{2} + 12(Y_{9}-8)^{2}-7Y_{10}\leq 0,\\ &\quad X_{i}=x_{i}+\frac{t}{100},\quad Y_{i}=x_{i}+sin \frac{k\pi t}{2},\quad -10-sin \frac{k\pi t}{2}\leq x_{i}\leq 10+sin \frac{k\pi t}{2},\quad 1\leq i\leq 10.\\ \end{aligned} $$
$$ \begin{aligned}&G08.\,Min\,f(X) = -\frac{sin^{3}(2\pi X_{1}) sin(2\pi X_{2})}{X^{3}_{1}(X_{1} + X_{2})}\\ &s.t.,\\&\quad g_{1}(Y) = Y^{2}_{1}-Y_{2} + 1\leq 0, \quad g_{2}(Y) = 1-Y_{1} + (Y_{2}-4)^{2}\leq 0,\\ &\quad X_{i}=\left(cos\frac{k\pi t}{2}\right)x_{i},\quad Y_{i}=x_{i},\quad -sin \frac{k\pi t}{2}\leq x_{1},\quad x_{2}\leq 10+sin \frac{k\pi t}{2}.\\ \end{aligned} $$
$$ \begin{aligned}&G09.\,Min\,f(X) = (X_{1}-10)^{2} + 5(X_{2}-12)^{2} + X^{4}_{3} + 3(X_{4}-11)^{2} +10X_{5}^{6} + 7X_{6}^{2}\\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+X_{7}^{4}-4X_{6}X_{7}-10X_{6}-8X_{7}\\ &s.t.,\\ &\quad g_{1}(Y) = -127 + 2Y^{2}_{1} + 3Y^{4}_{2} + Y_{3} + 4Y^{2}_{4} + 5Y_{5}\leq 0,\\ &\quad g_{2}(Y) = -282 + 7Y_{1} + 3Y_{2} + 10Y^{2}_{3} + Y_{4}-Y_{5}\leq 0,\\ &\quad g_{3}(Y) =-196 + 23Y_{1} + Y_{2}^{2} + 6Y^{2}_{6}-8Y_{7}\leq 0,\\ &\quad g_{4}(Y) = 4Y^{2}_{1} + Y_{2}^{2}-3Y_{1}Y_{2} + 2Y^{2}_{3} + 5Y_{6}-11Y_{7}\leq 0,\\ &\quad X_{i}=cos\frac{k\pi t}{2} \left(x_{i}+\frac{\pi t}{7}\right),\quad Y_{i}=x_{i},\quad -10-sin \frac{k\pi t}{2} \leq x_{i} \leq 10+sin \frac{k\pi t}{2}.\\ \end{aligned} $$
$$ \begin{aligned}& c02.\,Min\,f(X) = max(X_{1},X_{2},...,X_{p(t)})\\ &s.t.,\\ &\quad g_{1}(Y) = 10-\frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}[Y_{i}^{2}-10cos(2\pi Y_{i})+10]\leq 0,\\ &\quad g_{2}(Y) = \frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}[Y_{i}^{2}-10cos(2\pi Y_{i})+10]-15\leq 0,\\ &\quad g_{3}(Y) =\frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}[Y_{i}^{2}-10cos(2\pi Y_{i})+10]-20= 0,\\ &\quad X_{i}=\left(cos\frac{k \pi t}{100p(t)}\right)x_{i},\quad Y_{i}=x_{i},\quad -5.12-sin \frac{k\pi t}{2} \leq x_{i} \leq 5.12+sin \frac{k\pi t}{2},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10, \quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}& c03. \,Min\,f(X) = \sum\limits_{i=1}^{p(t)-1}(100(X_{i}^{2}-X_{i+1})^{2}+(X_{i}-1)^{2})\\ & s.t.,\\ &\quad h(Y) = \sum\limits_{i=1}^{p(t)-1}(Y_{i}-Y_{i+1})^{2}=0,\\ &\quad X_{i}=\left(1+sin\frac{k \pi t}{2}\right)\left(x_{i}+\frac{\pi t}{20}\right),\quad Y_{i}=\left(1+\frac{k\pi t}{20}\right)x_{i},\quad -1000-sin \frac{k\pi t}{2} \leq x_{i} \leq 1000+sin \frac{k\pi t}{2},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}&c04.\,Min\,f(X) = max(X_{1},X_{2},...,X_{p(t)})\\ & s.t.,\\ &\quad h_{1}(Y) = \frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}(Y_{i}cos\sqrt{|Y_{i}|})=0,\\&\quad h_{2}(Y) = \sum\limits_{i=1}^{p(t)/2-1}(Y_{i}-Y_{i+1})^{2}=0,\\ &\quad h_{3}(Y) = \sum\limits_{i=p(t)/2+1}^{p(t)-1}(Y_{i}^{2}-Y_{i+1})^{2}=0,\\ &\quad h_{4}(Y) = \sum\limits_{i=1}^{p(t)}Y_{i}=0,\\ &\quad X_{i}=x_{i}+\frac{\pi t}{3},\quad Y_{i}=\left(1+\frac{k\pi t}{20}\right)x_{i},\quad -50-sin \frac{k\pi t}{2} \leq x_{i} \leq 50+sin \frac{k\pi t}{2},\quad 1\leq i\leq p(t),\\ &\quad p(t)=5, \quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}&c05.\, Min\,f(X) = max(X_{1},X_{2},...,X_{p(t)})\\ &s.t.,\\ &\quad h_{1}(Y) = \frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}(-Y_{i}sin\sqrt{|Y_{i}|})=0,\\&\quad h_{2}(Y) = \frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}(-Y_{i}cos\sqrt{0.5|Y_{i}|})=0,\\&\quad X_{i}=cos\frac{k\pi t}{20}\left(x_{i}+sin\frac{\pi t}{20}\right),\quad Y_{i}=x_{i},\quad -600-sin \frac{k\pi t}{20} \leq x_{i} \leq 600+sin \frac{k\pi t}{20},\quad 1\leq i\leq p(t),\\&\quad p(t)=5, \quad 1\leq t\leq 5, \quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}&c07. \,Min\,f(X) = \sum\limits_{i=1}^{p(t)-1}(100(X_{i}^{2}-X_{i+1})^{2}+(X_{i}-1)^{2})\\ &s.t.,\\ &\quad g_{1}(Y) = 0.5-exp\left(-0.1\sqrt{\frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}Y_{i}^{2}}\right)-3exp\left(\frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}cos(0.1Y_{i})\right)+exp(1)\leq 0,\\ &\quad X_{i}=\left(cos\frac{k\pi t}{20p(t)}\right)x_{i},\quad Y_{i}=\left(cos\frac{k\pi t}{20p(t)}\right)x_{i},\quad -140-sin \frac{k\pi t}{20} \leq x_{i} \leq 140+sin \frac{k\pi t}{20},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}&c09.\, Min\,f(X) = \sum\limits_{i=1}^{p(t)-1}(100(X_{i}^{2}-X_{i+1})^{2}+(X_{i}-1)^{2})\\ &s.t.,\\ &\quad h(Y) = \sum\limits_{i=1}^{p(t)}Y_{i}sin \sqrt{|Y_{i}|}= 0,\\ &\quad X_{i}=\left(cos\frac{k\pi t}{2000}\right)x_{i},\quad Y_{i}=x_{i}+sin\left(\frac{\pi t}{2000}\right),\quad -500-sin \frac{k\pi t}{2000} \leq x_{i} \leq 500+sin \frac{k\pi t}{2000},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned} &c11.\, Min\,f(X) =- \frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}X_{i}cos(2\sqrt{|X_{i}|})\\ &s.t.,\\ &\quad h(Y) = \sum\limits_{i=1}^{p(t)-1}(100(Y_{i}^{2}-Y_{i+1})^{2}+(Y_{i}-1)^{2})= 0,\\ &\quad X_{i}=cos\frac{k\pi t}{2000p(t)}\left(x_{i}+sin\frac{k\pi t}{2000p(t)}\right),\quad Y_{i}=\left(cos\frac{k\pi t}{2000p(t)}\right) x_{i},\quad -100-sin \frac{k\pi t}{2000} \leq x_{i} \leq 100+sin \frac{k\pi t}{2000},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}&c12.\, Min\,f(X) = \sum\limits_{i=1}^{p(t)}X_{i}sin \sqrt{|X_{i}|}\\ &s.t.,\\ &\quad h(Y) = \sum\limits_{i=1}^{p(t)-1}(Y_{i}^{2}-Y_{i+1})^{2}= 0,\\ &\quad g(Y) = \sum\limits_{i=1}^{p(t)}(Y_{i}-100cos(0.1Y_{i})+10)\leq 0,\\ &\quad X_{i}=cos\frac{k\pi t}{2000}\left(x_{i}+sin\frac{\pi t}{2000}\right),\quad Y_{i}=x_{i}+sin\frac{\pi t}{2000},\quad -1000-sin \frac{k\pi t}{2000} \leq x_{i} \leq 1000+sin \frac{k\pi t}{2000},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned} &c13. \,Min\,f(X) = -\frac{1}{p(t)}\sum\limits_{i=1}^{p(t)}X_{i}sin(\sqrt{|X_{i}|})\\ &s.t.,\\ &\quad g_{1}(Y) = -50+\frac{1}{100p(t)} \sum\limits_{i=1}^{p(t)}Y_{i}^{2}\leq 0,\\ &\quad g_{2}(Y) = \frac{50}{p(t)} \sum\limits_{i=1}^{p(t)}sin\left(\frac{1}{50}\pi Y_{i}\right)\leq 0,\\ &\quad g_{3}(Y) = 76-50\left(\sum\limits_{i=1}^{p(t)}\frac{Y_{i}^{2}}{4000}-\prod\limits_{i=1}^{p(t)}cos \frac{Y_{i}}{\sqrt{i}}\right)\leq 0, \\ &\quad X_{i}=x_{i}+sin\frac{\pi t}{2000},\quad Y_{i}=cos\frac{k\pi t}{2000}\left(x_{i}+sin\frac{\pi t}{2000}\right),\quad -500-sin \frac{k\pi t}{2000} \leq x_{i} \leq 500+sin \frac{k\pi t}{2000},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}&c14.\,Min\,f(X) = \sum\limits_{i=1}^{p(t)-1}(100(X_{i}^{2}-X_{i+1})^{2}+(X_{i}-1)^{2})\\& s.t.,\\&\quad g_{1}(Y) = \sum\limits_{i=1}^{p(t)}(-Y_{i}cos\sqrt{|Y_{i}|})-p(t)\leq 0,\\&\quad g_{2}(Y) = \sum\limits_{i=1}^{p(t)}(Y_{i}cos\sqrt{|Y_{i}|})-p(t)\leq 0, \\ &\quad g_{3}(Y) = \sum\limits_{i=1}^{p(t)}(Y_{i}sin\sqrt{|Y_{i}|})-10p(t)\leq 0,\\ &\quad X_{i}=x_{i}+sin\frac{\pi t}{2000},\quad Y_{i}=x_{i},\quad -1000-sin \frac{k\pi t}{2} \leq x_{i} \leq 1000+sin \frac{k\pi t}{2},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}&c16.\, Min\,f(X) = \sum\limits_{i=1}^{p(t)}\frac{X_{i}^{2}}{4000}-\prod\limits_{i=1}^{p(t)}cos\left(\frac{Y_{i}}{\sqrt{i}}\right)+1\\ &s.t.,\\ &\quad g_{1}(Y) = \sum_{i=1}^{p(t)}[Y_{i}^{2}-100cos(\pi Y_{i}+10)]\leq 0,\\ &\quad g_{2}(Y) = \prod\limits_{i=1}^{p(t)}Y_{i}\leq 0,\\ &\quad h(Y) = \sum\limits_{i=1}^{p(t)}(Y_{i}sin\sqrt{|Y_{i}|})= 0,\\ &\quad X_{i}=x_{i},\quad Y_{i}=\left(1+sin\frac{k \pi t}{20}\right)x_{i},\quad -10-sin \frac{k\pi t}{2} \leq x_{i} \leq 10+sin \frac{k\pi t}{2},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}&c17. \,Min\,f(X) = \sum\limits_{i=1}^{p(t)}(X_{i}^{2}-X_{i+1})^{2}\\ &s.t.,\\ &\quad g(Y) = \prod\limits_{i=1}^{p(t)}Y_{i}\leq 0,\\ &\quad h(Y)=\sum\limits_{i=1}^{p(t)}X_{i}sin 4\sqrt{|Y_{i}|}=0, \\ &\quad X_{i}=x_{i},\quad Y_{i}=x_{i}+sin\frac{\pi t}{20},\quad -10-sin \frac{k\pi t}{2} \leq x_{i} \leq 10+sin \frac{k\pi t}{2},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$
$$ \begin{aligned}&c18.\, Min\,f(X) = \sum\limits_{i=1}^{p(t)}(X_{i}^{2}-X_{i+1})^{2}\\ &s.t.,\\ &\quad h(Y)=\sum\limits_{i=1}^{p(t)}Y_{i}sin \sqrt{|Y_{i}|}=0,\\ &\quad X_{i}=x_{i},\quad Y_{i}=\left(1+sin\frac{k\pi t}{20}\right)\left(x_{i}+sin\frac{\pi t}{200}\right),\quad -50-sin \frac{k\pi t}{2} \leq x_{i} \leq 50+sin \frac{k\pi t}{2},\\ &\quad p(t)=5,\quad 1\leq t\leq 5,\quad p(t)=10,\quad t>5.\\ \end{aligned} $$

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Zhang, Z., Yue, S., Liao, M. et al. Danger theory based artificial immune system solving dynamic constrained single-objective optimization. Soft Comput 18, 185–206 (2014). https://doi.org/10.1007/s00500-013-1048-0

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