Function |
Equation |
Dimension (n) |
Search Space (xmin,
xmax) |
Global Minimum |
Paraboloid |
\(f\left(\ x\ \right)=x_{1}^{2}+x_{2}^{2}\) |
2 |
[-100,100] |
0 |
Sphere |
\(f\left(\ x\ \right)=\sum_{i=1}^{n}x_{i}^{2}\) |
20,000 |
[-100, 100] |
0 |
Rosenbrock |
\(f\left(x\right)=\sum_{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left(x_{i}-1\right)^{2}\right]\) |
20,000 |
[-30, 30] |
0 |
Rastrigin |
\(f\left(x\right)=\sum_{i=1}^{n}\left[x_{i}^{2}-10\cos\left(2\pi x_{i}\right)+10\right]\) |
20,000 |
[-5.12, 5.12] |
0 |
Griewank |
\(f\left(x\right)=\frac{1}{4000}\sum_{i=1}^{n}\left[x_{i}^{2}-\prod_{i=1}^{n}{\cos\left(\frac{x_{i}}{\sqrt{i}}+1\right)}\right]\) |
20,000 |
[-600, 600] |
0 |
Shaffer f6 |
\(f\left(x\right)=0.5-\frac{\left(\sin\sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{2}-0.5}{\left[1+0.001\left(x_{1}^{2}+x_{2}^{2}\right)\right]^{2}}\) |
2 |
[-100, 100] |
0 |
Alpine |
\(f\left(x\right)=\sum_{i=1}^{n}\left[x_{i}\sin\left(x_{i}\right)+0.1x_{i}\right]\) |
20,000 |
[-10, 10] |
0 |
Brown |
\(f\left(x\right)=\sum_{i=1}^{n-1}\left[\left(x_{i}^{2}\right)^{x_{i+1}^{2}+1}+\left(x_{i+1}^{2}\right)^{x_{i}^{2}+1}\right]\) |
20,000 |
[-1, 1] |
0 |
Chung Reynolds |
\(f\left(x\right)=\left(\sum_{i=1}^{n}\left[x_{i}^{2}\right]\right)^{2}\) |
20,000 |
[-100, 100] |
0 |
Dixon Price |
\(f\left(x\right)=\left(x_{1}-1\right)^{2}+\sum_{i=2}^{n}\left[i\left(2x_{i}^{2}-x_{i-1}\right)^{2}\right]\) |
20,000 |
[-10, 10] |
0 |
Exponential |
\(f\left(x\right)=-exp\left(-0.5\sum_{i=1}^{n}\left[x_{i}^{2}\right]\right)\) |
20,000 |
[-1, 1] |
-1 |
Salomon |
\(f\left(x\right)=1-\cos\left(2\pi\sqrt{\sum_{i=1}^{n}\left[x_{i}^{2}\right]}\right)+0.1\sqrt{\sum_{i=1}^{n}\left[x_{i}^{2}\right]}\) |
20,000 |
[-100, 100] |
0 |
Schumer Steiglitz |
\(f\left(x\right)=\sum_{i=1}^{n}\left[x_{i}^{4}\right]\) |
20,000 |
[-100, 100] |
0 |
Sum of Powers |
\(f\left(x\right)=\sum_{i=1}^{n}\left[\left|x_{i}\right|^{i+1}\right]\) |
20,000 |
[-1, 1] |
0 |
Sum of Squares |
\(f\left(x\right)=\sum_{i=1}^{n}\left[ix_{i}^{2}\right]\) |
20,000 |
[-1, 1] |
0 |
Zakharov |
\(f\left(x\right)=\sum_{i=1}^{n}\left[x_{i}^{2}\right]+\left(\sum_{i=1}^{n}\left[0.5ix_{i}\right]\right)^{2}+\left(\sum_{i=1}^{n}\left[0.5ix_{i}\right]\right)^{4}\) |
20,000 |
[-10, 10] |
0 |