N-D Test Functions Z¶

class go_benchmark.Zacharov(dimensions=2)¶

Zacharov test objective function.

This class defines the Zacharov global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Zacharov}}(\mathbf{x}) = \sum_{i=1}^{n} x_i^2 + \left ( \frac{1}{2} \sum_{i=1}^{n} i x_i \right )^2 + \left ( \frac{1}{2} \sum_{i=1}^{n} i x_i \right )^4$

Here, $n$ represents the number of dimensions and $x_i \in [-5, 10]$ for $i=1,...,n$ .

Two-dimensional Zacharov function

Global optimum: $f(x_i) = 0$ for $x_i = 0$ for $i=1,...,n$

class go_benchmark.ZeroSum(dimensions=2)¶

ZeroSum test objective function.

This class defines the ZeroSum global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{ZeroSum}}(\mathbf{x}) = \begin{cases}0 & \textrm{if} \sum_{i=1}^n x_i = 0 \\ 1 + \left(10000 \left |\sum_{i=1}^n x_i\right| \right)^{0.5} & \textrm{otherwise}\end{cases}$

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i=1,...,n$ .

Two-dimensional ZeroSum function

Global optimum: $f(x_i) = 0$ where $\sum_{i=1}^n x_i = 0$

class go_benchmark.Zettl(dimensions=2)¶

Zettl test objective function.

This class defines the Zettl global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Zettl}}(\mathbf{x}) = \frac{1}{4} x_{1} + \left(x_{1}^{2} - 2 x_{1} + x_{2}^{2}\right)^{2}$

Here, $n$ represents the number of dimensions and $x_i \in [-1, 5]$ for $i=1,2$ .

Two-dimensional Zettl function

Global optimum: $f(x_i) = -0.0037912$ for $\mathbf{x} = [-0.029896, 0.0]$

class go_benchmark.Zimmerman(dimensions=2)¶

Zimmerman test objective function.

This class defines the Zimmerman global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Zimmerman}}(\mathbf{x}) = \max \left[Zh1(x), Zp(Zh2(x))\textrm{sgn}(Zh2(x)), Zp(Zh3(x)) \textrm{sgn}(Zh3(x)), Zp(-x_1)\textrm{sgn}(x_1), Zp(-x_2)\textrm{sgn}(x_2) \right]$