N-D Test Functions T¶

class go_benchmark.TestTubeHolder(dimensions=2)¶

TestTubeHolder test objective function.

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

$f_{\text{TestTubeHolder}}(\mathbf{x}) = - 4 \left | {e^{\left|{\cos\left(\frac{1}{200} x_{1}^{2} + \frac{1}{200} x_{2}^{2}\right)}\right|} \sin\left(x_{1}\right) \cos\left(x_{2}\right)}\right |$

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

Two-dimensional TestTubeHolder function

Global optimum: $f(x_i) = -10.872299901558$ for $\mathbf{x} = [-\pi/2, 0]$

class go_benchmark.ThreeHumpCamel(dimensions=2)¶

Three Hump Camel test objective function.

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

$f_{\text{ThreeHumpCamel}}(\mathbf{x}) = 2x_1^2 - 1.05x_1^4 + \frac{x_1^6}{6} + x_1x_2 + x_2^2$

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

Two-dimensional Three Hump Camel function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [0, 0]$

class go_benchmark.Treccani(dimensions=2)¶

Treccani test objective function.

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

$f_{\text{Treccani}}(\mathbf{x}) = x_1^4 + 4x_1^3 + 4x_1^2 + x_2^2$

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

Two-dimensional Treccani function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [-2, 0]$ or $\mathbf{x} = [0, 0]$ .

class go_benchmark.Trefethen(dimensions=2)¶

Trefethen test objective function.

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

$f_{\text{Trefethen}}(\mathbf{x}) = 0.25 x_{1}^{2} + 0.25 x_{2}^{2} + e^{\sin\left(50 x_{1}\right)} - \sin\left(10 x_{1} + 10 x_{2}\right) + \sin\left(60 e^{x_{2}}\right) + \sin\left[70 \sin\left(x_{1}\right)\right] + \sin\left[\sin\left(80 x_{2}\right)\right]$

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

Two-dimensional Trefethen function

Global optimum: $f(x_i) = -3.3068686474$ for $\mathbf{x} = [-0.02440307923, 0.2106124261]$

class go_benchmark.Trid(dimensions=6)¶

Trid test objective function.

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

$f_{\text{Trid}}(\mathbf{x}) = \sum_{i=1}^{n}(x_i - 1)^2 - \sum_{i=2}^{n} x_ix_{i-1}$

Here, $n$ represents the number of dimensions and $x_i \in [-20, 20]$ for $i=1,...,6$ .

Global optimum: $f(x_i) = -50$ for $\mathbf{x} = [6, 10, 12, 12, 10, 6]$

class go_benchmark.Trigonometric01(dimensions=2)¶

Trigonometric 1 test objective function.

This class defines the Trigonometric 1 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Trigonometric01}}(\mathbf{x}) = \sum_{i=1}^{n} \left [n - \sum_{j=1}^{n} \cos(x_j) + i \left(1 - cos(x_i) - sin(x_i) \right ) \right]^2$

Here, $n$ represents the number of dimensions and $x_i \in [0, \pi]$ for $i=1,...,n$ .

Two-dimensional Trigonometric 1 function

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

class go_benchmark.Trigonometric02(dimensions=2)¶

Trigonometric 2 test objective function.

This class defines the Trigonometric 2 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Trigonometric2}}(\mathbf{x}) = 1 + \sum_{i=1}^{n} 8 \sin^2 \left[7(x_i - 0.9)^2 \right] + 6 \sin^2 \left[14(x_i - 0.9)^2 \right] + (x_i - 0.9)^2$

Here, $n$ represents the number of dimensions and $x_i \in [-500, 500]$ for $i=1,...,n$ .

Two-dimensional Trigonometric 2 function

Global optimum: $f(x_i) = 1$ for $x_i = 0.9$ for $i=1,...,n$

class go_benchmark.Tripod(dimensions=2)¶

Tripod test objective function.

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

$f_{\text{Tripod}}(\mathbf{x}) = p(x_2) \left[1 + p(x_1) \right] + \lvert x_1 + 50p(x_2) \left[1 - 2p(x_1) \right] \rvert + \lvert x_2 + 50\left[1 - 2p(x_2)\right] \rvert$