N-D Test Functions Z¶

class Zacharov(dimensions=2)¶

Zacharov objective function.

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

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

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

Two-dimensional Zacharov function

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

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Zagros(dimensions=2)¶: Zagros objective function.

Two-dimensional Zagros function

class ZeroSum(dimensions=2)¶

ZeroSum objective function.

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

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

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

Two-dimensional ZeroSum function

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

Gavana, A. Global Optimization Benchmarks and AMPGO

class Zettl(dimensions=2)¶

Zettl objective function.

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

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

with $x_i \in [-1, 5]$ for $i = 1, 2$ .

Two-dimensional Zettl function

Global optimum: $f(x) = -0.0037912$ for $x = [-0.029896, 0.0]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Zimmerman(dimensions=2)¶

Zimmerman objective function.

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

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

Where, in this exercise:

$\begin{cases} Zh1(x) = 9 - x_1 - x_2 \\ Zh2(x) = (x_1 - 3)^2 + (x_2 - 2)^2 \\ Zh3(x) = x_1x_2 - 14 \\ Zp(t) = 100(1 + t) \\ \end{cases}$

Where $x$ is a vector and $t$ is a scalar.

Here, $x_i \in [0, 100]$ for $i = 1, 2$ .

Two-dimensional Zimmerman function

Global optimum: $f(x) = 0$ for $x = [7, 2]$

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

implementation from Gavana

class Zirilli(dimensions=2)¶

Zettl objective function.

This class defines the Zirilli global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{Zirilli}}(x) = 0.25x_1^4 - 0.5x_1^2 + 0.1x_1 + 0.5x_2^2$

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i = 1, 2$ .

Two-dimensional Zirilli function

Global optimum: $f(x) = -0.3523$ for $x = [-1.0465, 0]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

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