Table 1 First part of benchmark functions
From: An accelerated and robust algorithm for ant colony optimization in continuous functions
Function | Formula | Optimal x* | Minimum f(x*) |
|---|---|---|---|
Sphere | \( f\left(\overrightarrow{x}\right)=\sum \limits_{i=1}^n{x_i}^2 \) | \( \overrightarrow{x^{\ast }}=\left(0,\dots, 0\right) \) | fmin = 0 |
Ellipsoid | \( f\left(\overrightarrow{x}\right)=\sum \limits_{i=1}^n{\left({100}^{\frac{i-1}{n-1}}{x}_i\right)}^2 \) | \( \overrightarrow{x^{\ast }}=\left(0,\dots, 0\right) \) | fmin = 0 |
Cigar | \( f\left(\overrightarrow{x}\right)={x_1}^2+{10}^4\sum \limits_{i=2}^n{x_i}^2 \) | \( \overrightarrow{x^{\ast }}=\left(0,\dots, 0\right) \) | fmin = 0 |
Tablet | \( f\left(\overrightarrow{x}\right)={10}^4{x_1}^2+\sum \limits_{i=2}^n{x_i}^2 \) | \( \overrightarrow{x^{\ast }}=\left(0,\dots, 0\right) \) | fmin = 0 |
Rosenbrock | \( f\left(\overrightarrow{x}\right)=\sum \limits_{i=1}^{n-1}\left[100{\left({x_i}^2-{x}_{i+1}\right)}^2+{\left({x}_i-1\right)}^2\right] \) | \( \overrightarrow{x^{\ast }}=\left(1,\dots, 1\right) \) | fmin = 0 |