Table 8 CEC 2019 100-Digit Challenge benchmark functions

Storn’s Chebyshev Polynomial Fitting Problem

\( f\left(\overrightarrow{x}\right)={p}_1+{p}_2+{p}_3 \)

\( {p}_1=\left\{\begin{array}{c}{\left(u-d\right)}^2\kern1.5em , if\ u<d\\ {}\kern1em 0,\kern4.75em , otherwise\end{array}\right. \)

\( u=\sum \limits_{j=1}^D{x}_j{(1.2)}^{D-j} \)

\( {p}_2=\left\{\begin{array}{c}{\left(v-d\right)}^2\kern1.5em , if\ v<d\\ {}\kern1em 0,\kern4.75em , otherwise\end{array}\right. \)

\( v=\sum \limits_{j=1}^D{x}_j{\left(-1.2\right)}^{D-j} \)

\( pk=\left\{\begin{array}{c}{\left({w}_k-1\right)}^2\kern2.5em , if\ {w}_k>1\\ {}{\left({w}_k+1\right)}^2\kern2.25em , if\ {w}_k<1\\ {}\kern1em 0,\kern5.5em , otherwise\end{array}\right. \)

\( {w}_k=\sum \limits_{j=1}^D{x}_j{\left(\frac{2k}{m}-1\right)}^{D-j} \)

\( {p}_3=\sum \limits_{m=0}^m{p}_k,\kern0.5em k=0,1,\dots, m,\kern0.5em m=32D \)

d = 72.661 for D = 9

f_min = 1

Inverse Hilbert Matrix Problem

\( f\left(\overrightarrow{x}\right)=\sum \limits_{i=1}^n\sum \limits_{k=1}^n\left|{w}_{i,k}\right| \)

\( \left({w}_{i,k}\right)=W= HZ-I,\kern0.5em I=\left[\begin{array}{cccc}1& 0& \cdots & 0\\ {}0& 1& \dots & 0\\ {}\vdots & 0& \ddots & \vdots \\ {}0& 0& \cdots & 1\end{array}\right] \)

\( H=\left({h}_{i,k}\right),\kern0.5em {h}_{i,k}=\frac{1}{i+k-1},\kern0.5em i,k=1,2,\dots, n,\kern0.5em n=\sqrt{D} \)

Z = (z_{i, k}), z_{i, k} = x_{i + n(k − 1)}

f_min =  − 186.7309

Lennard-Jones Minimum Energy Cluster

\( f\left(\overrightarrow{x}\right)=12.7120622568+\sum \limits_{i=1}^{n-1}\sum \limits_{j=i+1}^n\left(\frac{1}{d_{i,j}^2}-\frac{2}{d_{i,j}}\right) \)

\( {d}_{i,j}={\left(\sum \limits_{k=0}^2{\left({x}_{3i+k-2}-{x}_{3j+k-2}\right)}^2\right)}^3\kern1.5em ,n=\frac{D}{3} \)

f_min = 1

Rastrigin’s Function

\( f\left(\overrightarrow{x}\right)=\sum \limits_{i=1}^D\left({x}_i^2-10\cos \left(2\pi {x}_i\right)+10\right) \)

f_min = 1

Griewangk’s Function

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

f_min = 1

Weierstrass Function

\( f\left(\overrightarrow{x}\right)=\sum \limits_{i=1}^D\left(\sum \limits_{k=0}^{k_{max}}\left[{a}^k\cos \left(2\pi {b}^k\left({x}_i+0.5\right)\right)\right]\right)-D\sum \limits_{k=0}^{k_{max}}{a}^k\cos \left(\pi {b}^k\right) \)

a = 0.5, b = 3,  k_max = 20

f_min = 1

Modified Schwefel’s Function

\( f\left(\overrightarrow{x}\right)=418.9829D-\sum \limits_{i=1}^Dg\left({z}_i\right) \)

z_i = x_i + 420.9687462275036

\( g\left({z}_i\right)=\left\{\begin{array}{c}{z}_i\sin \left({\left|{z}_i\right|}^{\frac{1}{2}}\right)\kern19.60em if\mid {z}_i\mid \le 500\\ {}\left(500-\mathit{\operatorname{mod}}\left({z}_i,500\right)\right)\sin \left(\sqrt{\left|500-\mathit{\operatorname{mod}}\left({z}_i,500\right)\right|}\right)-\frac{{\left({z}_i-500\right)}^2}{10000D}\kern2.29em if\ {z}_i>500\ \\ {}\left(\mathit{\operatorname{mod}}\left({z}_i,500\right)-500\right)\sin \left(\sqrt{\left|\mathit{\operatorname{mod}}\left({z}_i,500\right)-500\right|}\right)-\frac{{\left({z}_i+500\right)}^2}{10000D}\kern1.55em if\ {z}_i<-500\ \end{array}\right. \)

f_min = 1

Expanded Schaffer’s F6 Function

\( g\left(x,y\right)=0.5+\frac{\sin^2\left(\sqrt{x^2+{y}^2}\right)-0.5}{\left(1+0.001\Big({x}^2+{y}^2\right)\Big){}^2} \)

\( f\left(\overrightarrow{x}\right)=g\left({x}_1,{x}_2\right)+g\left({x}_2,{x}_3\right)\dots +g\left({x}_{D-1},{x}_D\right)+g\left({x}_D,{x}_1\right) \)

f_min = 1

Happy Cat Function

\( f\left(\overrightarrow{x}\right)={\left|\sum \limits_{i=1}^D{x}_i^2-D\right|}^{1/4}+\left(0.5\sum \limits_{i=1}^D{x}_i^2+\sum \limits_{i=1}^D{x}_i\right)/D+0.5 \)

f_min = 1

Ackley Function

\( f\left(\overrightarrow{x}\right)=-20\exp \left(-0.2\sqrt{\frac{1}{n}\sum \limits_{i=1}^n{x_i}^2}\right)-\exp \left(\frac{1}{n}\sum \limits_{i=1}^n\cos \left(2\pi {x}_i\right)\right) \)

f_min = 1