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

de Freitas, Jairo G.; Yamanaka, Keiji

doi:10.1186/s13173-021-00116-8

Journal of the Brazilian Computer Society

Table 6 Third part of benchmark functions

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

Function	Formula	Optimal x*	Minimum f(x*)
Rastrigin	$ f\left(\overrightarrow{x}\right)=\Big({x_1}^2+{x_2}^2-\cos \left(18{x}_1\right)-\cos 18{x}_2 $	$ \overrightarrow{x^{\ast }}=\left(0,0\right) $	f_min = 0.397887
Shubert	$ f\left(\overrightarrow{x}\right)=\left(\sum \limits_{i=1}^5i\cos \left(i+\left(i+1\right){x}_1\right)\right)\left(\sum \limits_{i=1}^5i\cos \left(i+\left(i+1\right){x}_2\right)\right) $	18 optima	f_min = − 186.7309
Ackley	$ f\left(\overrightarrow{x}\right)=-20\exp \left(-0.2\sqrt{\frac{1}{n}\sum \limits_{i=1}^n{x_i}^2}\right)-\exp \left(\frac{1}{n}\sum \limits_{i=1}^n\cos \left(2\pi {x}_i\right)\right) $	$ \overrightarrow{x^{\ast }}=\left(0,\dots, 0\right) $	f_min = 0
Levy	$ f\left(\overrightarrow{x}\right)={\sin}^2\left(\pi {y}_1\right)+\sum \limits_{i=1}^{n-1}\left[{\left({y}_i-1\right)}^2\left(1+10{\sin}^2\left({\pi y}_1+1\right)\right)\right]+{\left({y}_n-1\right)}^2\left(1+10{\sin}^22\pi {y}_n\right)\Big) $ $ {y}_i=1+\frac{1}{4}\left({x}_i-1\right) $	$ \overrightarrow{x^{\ast }}=\left(1,\dots, 1\right) $	f_min = 0
Beale	$ f\left(\overrightarrow{x}\right)={\left(1.5-{x}_1+{x}_1{x}_2\right)}^2+{\left(2.25-{x}_1+{x}_1{x_2}^2\right)}^2+{\left(2.625-{x}_1+{x}_1{x_2}^3\right)}^2 $	$ \overrightarrow{x^{\ast }}=\left(3,0.5\right) $	f_min = 0
Six-Hump Camel-Back	$ f\left(\overrightarrow{x}\right)=4{x_1}^2-2.1{x_1}^4+\raisebox{1ex}{${x_1}^6$}\!\left/ \!\raisebox{-1ex}{$3$}\right.+{x}_1{x}_2-4{x_2}^2+4{x_2}^4 $	$ \overrightarrow{x^{\ast }}=\left(0.08983,-0.7126\right), $ ( − 0.08983, 0.7126	f_min = 0
Bohachevsky	$ f\left(\overrightarrow{x}\right)={x_1}^2+{x_2}^2-0.3\cos \left(3\pi {x}_1\right)+0.3\cos \left(4\pi {x}_2\right)+0.3 $	$ \overrightarrow{x^{\ast }}=\left(0,0.24\right),\left(0,-0.24\right) $	f_min = − 0.24
Hansen	$ f\left(\overrightarrow{x}\right)=\left(\cos (1)+2\cos \left({x}_1+2\right)+3\cos \left(2{x}_1+3\right)+4\cos \left(3{x}_1+4\right)+5\cos \left(4{x}_1+5\right)\right)\left(\cos 2{x}_2+1\right)+2\cos \left(3{x}_2+2\right)+3\cos \left(4{x}_2+3\right)+4\cos 5{x}_2+4\left)+5\cos \left(6{x}_2+5\right)\right) $	$ \overrightarrow{x^{\ast }}=\left(-1.30671,4.85806\right) $	f_min = − 176.5418

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Function	Formula	Optimal x*	Minimum f(x*)
Rastrigin	\( f\left(\overrightarrow{x}\right)=\Big({x_1}^2+{x_2}^2-\cos \left(18{x}_1\right)-\cos 18{x}_2 \)	\( \overrightarrow{x^{\ast }}=\left(0,0\right) \)	f_min = 0.397887
Shubert	\( f\left(\overrightarrow{x}\right)=\left(\sum \limits_{i=1}^5i\cos \left(i+\left(i+1\right){x}_1\right)\right)\left(\sum \limits_{i=1}^5i\cos \left(i+\left(i+1\right){x}_2\right)\right) \)	18 optima	f_min = − 186.7309
Ackley	\( f\left(\overrightarrow{x}\right)=-20\exp \left(-0.2\sqrt{\frac{1}{n}\sum \limits_{i=1}^n{x_i}^2}\right)-\exp \left(\frac{1}{n}\sum \limits_{i=1}^n\cos \left(2\pi {x}_i\right)\right) \)	\( \overrightarrow{x^{\ast }}=\left(0,\dots, 0\right) \)	f_min = 0
Levy	\( f\left(\overrightarrow{x}\right)={\sin}^2\left(\pi {y}_1\right)+\sum \limits_{i=1}^{n-1}\left[{\left({y}_i-1\right)}^2\left(1+10{\sin}^2\left({\pi y}_1+1\right)\right)\right]+{\left({y}_n-1\right)}^2\left(1+10{\sin}^22\pi {y}_n\right)\Big) \) \( {y}_i=1+\frac{1}{4}\left({x}_i-1\right) \)	\( \overrightarrow{x^{\ast }}=\left(1,\dots, 1\right) \)	f_min = 0
Beale	\( f\left(\overrightarrow{x}\right)={\left(1.5-{x}_1+{x}_1{x}_2\right)}^2+{\left(2.25-{x}_1+{x}_1{x_2}^2\right)}^2+{\left(2.625-{x}_1+{x}_1{x_2}^3\right)}^2 \)	\( \overrightarrow{x^{\ast }}=\left(3,0.5\right) \)	f_min = 0
Six-Hump Camel-Back	\( f\left(\overrightarrow{x}\right)=4{x_1}^2-2.1{x_1}^4+\raisebox{1ex}{${x_1}^6$}\!\left/ \!\raisebox{-1ex}{$3$}\right.+{x}_1{x}_2-4{x_2}^2+4{x_2}^4 \)	\( \overrightarrow{x^{\ast }}=\left(0.08983,-0.7126\right), \) ( − 0.08983, 0.7126	f_min = 0
Bohachevsky	\( f\left(\overrightarrow{x}\right)={x_1}^2+{x_2}^2-0.3\cos \left(3\pi {x}_1\right)+0.3\cos \left(4\pi {x}_2\right)+0.3 \)	\( \overrightarrow{x^{\ast }}=\left(0,0.24\right),\left(0,-0.24\right) \)	f_min = − 0.24
Hansen	\( f\left(\overrightarrow{x}\right)=\left(\cos (1)+2\cos \left({x}_1+2\right)+3\cos \left(2{x}_1+3\right)+4\cos \left(3{x}_1+4\right)+5\cos \left(4{x}_1+5\right)\right)\left(\cos 2{x}_2+1\right)+2\cos \left(3{x}_2+2\right)+3\cos \left(4{x}_2+3\right)+4\cos 5{x}_2+4\left)+5\cos \left(6{x}_2+5\right)\right) \)	\( \overrightarrow{x^{\ast }}=\left(-1.30671,4.85806\right) \)	f_min = − 176.5418