N-D Test Functions P¶

class go_benchmark.Parsopoulos(dimensions=2)¶

Parsopoulos test objective function.

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

$f_{\text{Parsopoulos}}(\mathbf{x}) = \cos(x_1)^2 + \sin(x_2)^2$

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

Two-dimensional Parsopoulos function

Global optimum: This function has inﬁnite number of global minima in R2, at points $\left(k\frac{\pi}{2}, \lambda \pi \right)$ , where $k = \pm1, \pm3, ...$ and $\lambda = 0, \pm1, \pm2, ...$

In the given domain problem, function has 12 global minima all equal to zero.

class go_benchmark.Pathological(dimensions=2)¶

Pathological test objective function.

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

$f_{\text{Pathological}}(\mathbf{x}) = \sum_{i=1}^{n -1} \frac{\sin^{2}\left(\sqrt{100 x_{i+1}^{2} + x_{i}^{2}}\right) -0.5}{0.001 \left(x_{i} - x_{i+1}\right)^{4} + 0.50}$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i=1,2$ .

Two-dimensional Pathological function

Global optimum: $f(x_i) = -1.99600798403$ for $x_i = 0$ for $i=1,2$

class go_benchmark.Paviani(dimensions=10)¶

Paviani test objective function.

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

$f_{\text{Paviani}}(\mathbf{x}) = \sum_{i=1}^{10} \left[\log^{2}\left(10 - x_i\right) + \log^{2}\left(x_i -2\right)\right] - \left(\prod_{i=1}^{10} x_i^{10} \right)^{0.2}$

Here, $n$ represents the number of dimensions and $x_i \in [2.001, 9.999]$ for $i=1,...,n$ .

Global optimum: $f(x_i) = -45.7784684040686$ for $x_i = 9.350266$ for $i=1,...,n$

class go_benchmark.Penalty01(dimensions=2)¶

Penalty 1 test objective function.

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

$f_{\text{Penalty01}}(\mathbf{x}) = \frac{\pi}{30} \left\{10 \sin^2(\pi y_1) + \sum_{i=1}^{n-1} (y_i - 1)^2 \left[1 + 10 \sin^2(\pi y_{i+1}) \right ] + (y_n - 1)^2 \right \} + \sum_{i=1}^n u(x_i, 10, 100, 4)$

Where, in this exercise:

$y_i = 1 + \frac{1}{4}(x_i + 1)$

And:

$u(x_i, a, k, m) = \begin{cases} k(x_i - a)^m & \textrm{if} \hspace{5pt} x_i > a \\ 0 & \textrm{if} \hspace{5pt} -a \leq x_i \leq a \\ k(-x_i - a)^m & \textrm{if} \hspace{5pt} x_i < -a \end{cases}$

Here, $n$ represents the number of dimensions and $x_i \in [-50, 50]$ for $i=1,...,n$ .

Two-dimensional Penalty 1 function

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

class go_benchmark.Penalty02(dimensions=2)¶

Penalty 2 test objective function.

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

$f_{\text{Penalty02}}(\mathbf{x}) = 0.1 \left\{\sin^2(3\pi x_1) + \sum_{i=1}^{n-1} (x_i - 1)^2 \left[1 + \sin^2(3\pi x_{i+1}) \right ] + (x_n - 1)^2 \left [1 + \sin^2(2 \pi x_n) \right ]\right \} + \sum_{i=1}^n u(x_i, 5, 100, 4)$

Where, in this exercise:

$u(x_i, a, k, m) = \begin{cases} k(x_i - a)^m & \textrm{if} \hspace{5pt} x_i > a \\ 0 & \textrm{if} \hspace{5pt} -a \leq x_i \leq a \\ k(-x_i - a)^m & \textrm{if} \hspace{5pt} x_i < -a \end{cases}$

Here, $n$ represents the number of dimensions and $x_i \in [-50, 50]$ for $i=1,...,n$ .

Two-dimensional Penalty 2 function

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

class go_benchmark.PenHolder(dimensions=2)¶

PenHolder test objective function.

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

$f_{\text{PenHolder}}(\mathbf{x}) = -e^{\left|{e^{\left|{- \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{\pi} + 1}\right|} \cos\left(x_{1}\right) \cos\left(x_{2}\right)}\right|^{-1}}$

Here, $n$ represents the number of dimensions and $x_i \in [-11, 11]$ for $i=1,2$ .

Two-dimensional PenHolder function

Global optimum: $f(x_i) = -0.9635348327265058$ for $x_i = \pm 9.646167671043401$ for $i=1,2$

class go_benchmark.PermFunction01(dimensions=2)¶

PermFunction 1 test objective function.

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

$f_{\text{PermFunction01}}(\mathbf{x}) = \sum_{k=1}^n \left\{ \sum_{j=1}^n (j^k + \beta) \left[ \left(\frac{x_j}{j}\right)^k - 1 \right] \right\}^2$

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

Two-dimensional PermFunction 1 function

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

class go_benchmark.PermFunction02(dimensions=2)¶

PermFunction 2 test objective function.

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

$f_{\text{PermFunction02}}(\mathbf{x}) = \sum_{k=1}^n \left\{ \sum_{j=1}^n (j + \beta) \left[ \left(x_j^k - \frac{1}{j} \right ) \right] \right\}^2$

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

Two-dimensional PermFunction 2 function

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

class go_benchmark.Pinter(dimensions=2)¶

Pinter test objective function.

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

$f_{\text{Pinter}}(\mathbf{x}) = \sum_{i=1}^n ix_i^2 + \sum_{i=1}^n 20i \sin^2 A + \sum_{i=1}^n i \log_{10} (1 + iB^2)$

Where, in this exercise:

$\begin{cases} A = x_{i-1} \sin x_i + \sin x_{i+1} \\ B = x_{i-1}^2 - 2x_i + 3x_{i+1} - \cos x_i + 1 \end{cases}$

Where $x_0 = x_n$ and $x_{n+1} = x_1$ .

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

Two-dimensional Pinter function

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

class go_benchmark.Plateau(dimensions=2)¶

Plateau test objective function.

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

$f_{\text{Plateau}}(\mathbf{x}) = 30 + \sum_{i=1}^n \lfloor x_i \rfloor$

Here, $n$ represents the number of dimensions and $x_i \in [-5.12, 5.12]$ for $i=1,...,n$ .

Two-dimensional Plateau function

Global optimum: $f(x_i) = 30$ for $x_i = 0$ for $i=1,...,n$

class go_benchmark.Powell(dimensions=4)¶

Powell test objective function.

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

$f_{\text{Powell}}(\mathbf{x}) = (x_3+10x_1)^2+5(x_2-x_4)^2+(x_1-2x_2)^4+10(x_3-x_4)^4$

Here, $n$ represents the number of dimensions and $x_i \in [-4, 5]$ for $i=1,...,4$ .

Global optimum: $f(x_i) = 0$ for $x_i = 0$ for $i=1,...,4$

class go_benchmark.PowerSum(dimensions=4)¶

Power sum test objective function.

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

$f_{\text{PowerSum}}(\mathbf{x}) = \sum_{k=1}^n\left[\left(\sum_{i=1}^n x_i^k \right) - b_k \right]^2$

Where, in this exercise, $\mathbf{b} = [8, 18, 44, 114]$

Here, $n$ represents the number of dimensions and $x_i \in [0, 4]$ for $i=1,...,4$ .

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [1, 2, 2, 3]$

class go_benchmark.Price01(dimensions=2)¶

Price 1 test objective function.

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

$f_{\text{Price01}}(\mathbf{x}) = (\lvert x_1 \rvert - 5)^2 + (\lvert x_2 \rvert - 5)^2$

Here, $n$ represents the number of dimensions and $x_i \in [-500, 500]$ for $i=1,2$ .

Two-dimensional Price 1 function

Global optimum: $f(x_i) = 0.0$ for $\mathbf{x} = [5, 5]$ or $\mathbf{x} = [5, -5]$ or $\mathbf{x} = [-5, 5]$ or $\mathbf{x} = [-5, -5]$

class go_benchmark.Price02(dimensions=2)¶

Price 2 test objective function.

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

$f_{\text{Price02}}(\mathbf{x}) = 1 + \sin^2(x_1) + \sin^2(x_2) - 0.1e^{(-x_1^2 - x_2^2)}$

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

Two-dimensional Price 2 function

Global optimum: $f(x_i) = 0.9$ for $x_i = 0$ for $i=1,2$

class go_benchmark.Price03(dimensions=2)¶

Price 3 test objective function.

This class defines the Price 3 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Price03}}(\mathbf{x}) = 100(x_2 - x_1^2)^2 + \left[6.4(x_2 - 0.5)^2 - x_1 - 0.6 \right]^2$

Here, $n$ represents the number of dimensions and $x_i \in [-50, 50]$ for $i=1,2$ .

Two-dimensional Price 3 function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [-5, -5]$ , $\mathbf{x} = [-5, 5]$ , $\mathbf{x} = [5, -5]$ , $\mathbf{x} = [5, 5]$

class go_benchmark.Price04(dimensions=2)¶

Price 4 test objective function.

This class defines the Price 4 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Price04}}(\mathbf{x}) = (2x_1^3x_2 - x_2^3)^2 + (6x_1 - x_2^2 + x_2)^2$

Here, $n$ represents the number of dimensions and $x_i \in [-50, 50]$ for $i=1,2$ .

Two-dimensional Price 4 function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [0, 0]$ , $\mathbf{x} = [2, 4]$ and $\mathbf{x} = [1.464, -2.506]$

N-D Test Functions P¶

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