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Table 3 Second part of benchmark functions

From: An accelerated and robust algorithm for ant colony optimization in continuous functions

Function

Formula

Optimal x*

Minimum f(x*)

Branin RCOS

\( f\left(\overrightarrow{x}\right)={\left({x}_2-\frac{5{x_1}^2}{4{\pi}^2}+\frac{5{x}_1}{\pi }-6\right)}^2+10\left(1-\frac{1}{8\pi }\ \right)\cos \left({x}_1\right)+10 \)

3 optimums

fmin = 0.397887

B2

\( f\left(\overrightarrow{x}\right)={x_1}^2+2{x_2}^2-0.3\cos \left(3\pi {x}_1\right)-0.4\cos \left(4\pi {x}_2\right)+0.7 \)

\( \overrightarrow{x^{\ast }}=\left(0,0\right) \)

fmin = 0

Easom

\( f\left(\overrightarrow{x}\right)=-\cos \left({x}_1\right)\cos \left({x}_2\right)\exp \left(-\left({\left({x}_1-\pi \right)}^2+{\left({x}_2-\pi \right)}^2\right)\right) \)

\( \overrightarrow{x^{\ast }}=\left(\pi, \pi \right) \)

fmin =  − 1

Goldstein and Price

\( f\left(\overrightarrow{x}\right)=\left[1+{\left({x}_1+{x}_2+1\right)}^2\left(19-14{x}_1+3{x_1}^2-14{x}_2+6{x}_1{x}_2\right)\right]\left[30+{\left(2{x}_1-3{x}_2\right)}^2\left(18-32{x}_1+12{x_1}^2+48{x}_2-36{x}_1{x}_2+27{x_2}^2\right)\right] \)

\( \overrightarrow{x^{\ast }}=\left(0,-1\right) \)

fmin = 3

Zakharov

\( f\left(\overrightarrow{x}\right)=\sum \limits_{i=1}^n{x_i}^2+{\left(\sum \limits_{i=1}^n0.5i{x}_i\right)}^2+{\left(\sum \limits_{i=1}^n0.5{ix}_i\right)}^4 \)

\( \overrightarrow{x^{\ast }}=\left(0,\dots, 0\right) \)

fmin = 0

De Jong

\( f\left(\overrightarrow{x}\right)={x_1}^2+{x_2}^2 \)+ x32

\( \overrightarrow{x^{\ast }}=\left(0,0,0\right) \)

fmin = 0

Hartmann (H3,4)

\( f\left(\overrightarrow{x}\right)=-\sum \limits_{i=1}^4{c}_i\exp \left(-\sum \limits_{j=1}^3{a}_{ij}{\left({x}_j-{p}_{ij}\right)}^2\right) \)

\( {a}_{ij}=\left|\begin{array}{ccc}3.0& 10.0& 30.0\\ {}0.1& 10.0& 35.0\\ {}3.0& 10.0& 30.0\\ {}0.1& 10.0& 35.0\end{array}\right| \)

\( {c}_i=\left|\begin{array}{c}1.0\\ {}1.2\\ {}3.0\\ {}3.2\end{array}\right| \)

\( {p}_{ij}=\left|\begin{array}{ccc}0.3689& 0.1170& 0.2673\\ {}0.4699& 0.4387& 0.7470\\ {}0.1091& 0.8732& 0.5547\\ {}0.0382& 0.5743& 0.8828\end{array}\right| \)

x1 = 0.114

x2 = 0.555

x3 = 0.855

fmin =  − 3.8628

Shekel (S4,k, k = 5, 7, 10)

\( f\left(\overrightarrow{x}\right)=-\sum \limits_{i=1}^k{\left(\sum \limits_{j=1}^4{\left({x}_j-{a}_{ji}\right)}^2+{c}_i\right)}^{-1} \)

\( {a}_{ij}=\left|\begin{array}{cccc}4.0& 4.0& 4.0& 4.0\\ {}1.0& 1.0& 1.0& 1.0\\ {}8.0& 8.0& 8.0& 8.0\\ {}6.0& 6.0& 6.0& 6.0\\ {}3.0& 7.0& 3.0& 7.0\\ {}2.0& 9.0& 2.0& 9.0\\ {}5.0& 3.0& 5.0& 3.0\\ {}8.0& 1.0& 8.0& 1.0\\ {}6.0& 2.0& 6.0& 2.0\\ {}7.0& 3.6& 7.0& 3.6\end{array}\right| \)

\( {c}_i=\left|\begin{array}{c}0.1\\ {}0.2\\ {}0.2\\ {}0.4\\ {}0.4\\ {}0.6\\ {}0.3\\ {}0.7\\ {}0.5\\ {}0.5\end{array}\right| \)

x1 = 4

x2 = 4

x3 = 4

x4 = 4

\( {f}_{min}^{k=5}=-10.1532 \)

\( {f}_{min}^{k=7}=-10.4029 \)

\( {f}_{min}^{k=10}=-10.5364 \)

Hartmann (H6,4)

\( f\left(\overrightarrow{x}\right)=-\sum \limits_{i=1}^4{c}_i\exp \left(-\sum \limits_{j=1}^3{a}_{ij}{\left({x}_j-{p}_{ij}\right)}^2\right) \)

\( {a}_{ij}=\left|\begin{array}{cccccc}10.0& 3.00& 17.0& 3.50& 1.70& 8.00\\ {}0.05& 10.0& 17.0& 0.10& 8.00& 14.0\\ {}3.00& 3.50& 1.70& 10.0& 17.0& 8.00\\ {}17.0& 8.00& 0.05& 10.0& 0.10& 14.0\end{array}\right| \)

\( {c}_i=\left|\begin{array}{c}1.0\\ {}1.2\\ {}3.0\\ {}3.2\end{array}\right| \)

\( {p}_{ij}=\left|\begin{array}{cccccc}0.1312& 0.1696& 0.5569& 0.0124& 0.8283& 0.5886\\ {}0.2329& 0.4135& 0.8307& 0.3736& 0.1004& 0.9991\\ {}0.2348& 0.1451& 0.3522& 0.2883& 0.3047& 0.6650\\ {}0.4047& 0.8828& 0.8732& 0.5743& 0.1091& 0.0381\end{array}\right| \)

x1 = 0.201

x2 = 0.150

x3 = 0.477

x4 = 0.275

x5 = 0.311

x6 = 0.657

fmin =  − 3.3223

Griewangk

\( f\left(\overrightarrow{x}\right)=\sum \limits_{i=1}^n\frac{{x_i}^2}{4000}-\prod \limits_{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)+1 \)

\( \overrightarrow{x^{\ast }}=\left(0,\dots, 0\right) \)

fmin = 0

Martin & Gaddy

\( f\left(\overrightarrow{x}\right)={\left({x}_1-{x}_2\right)}^2+{\left(\frac{x_1+{x}_2-10}{3}\right)}^2 \)

\( \overrightarrow{x^{\ast }}=\left(5,5\right) \)

fmin = 0