N-D Test Functions P¶

class Parsopoulos(dimensions=2)¶

Parsopoulos objective function.

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

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

with $x_i \in [-5, 5]$ 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.

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 Pathological(dimensions=2)¶

Pathological objective function.

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

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

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

Two-dimensional Pathological function

Global optimum: $f(x) = 0.$ for $x = [0, 0]$ for $i = 1, 2$

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 Paviani(dimensions=10)¶

Paviani objective function.

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

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

with $x_i \in [2.001, 9.999]$ for $i = 1, ... , 10$ .

Global optimum: $f(x_i) = -45.7784684040686$ for $x_i = 9.350266$ for $i = 1, ..., 10$

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.

Todo

think Gavana web/code definition is wrong because final product term shouldn’t raise x to power 10.

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

Two-dimensional Peaks function

class Penalty01(dimensions=2)¶

Penalty 1 objective function.

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

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

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 Penalty01 function

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

Gavana, A. Global Optimization Benchmarks and AMPGO

class Penalty02(dimensions=2)¶

Penalty 2 objective function.

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

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

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 Penalty02 function

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

Gavana, A. Global Optimization Benchmarks and AMPGO

class PenHolder(dimensions=2)¶

PenHolder objective function.

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

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

with $x_i \in [-11, 11]$ for $i = 1, 2$ .

Two-dimensional PenHolder function

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

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 PermFunction01(dimensions=2)¶

PermFunction 1 objective function.

This class defines the PermFunction1 global optimization problem. This is a multimodal minimization problem defined as follows:

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

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

Two-dimensional PermFunction01 function

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

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

Todo

line 560

class PermFunction02(dimensions=2)¶

PermFunction 2 objective function.

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

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

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

Two-dimensional PermFunction02 function

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

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

Todo

line 582

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

Two-dimensional Picheny function

class Pinter(dimensions=2)¶

Pinter objective function.

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

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

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: $f(x) = 0$ 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 Plateau(dimensions=2)¶

Plateau objective function.

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

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

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

Gavana, A. Global Optimization Benchmarks and AMPGO

class Powell(dimensions=4)¶

Powell objective function.

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

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

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

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

Powell, M. An iterative method for finding stationary values of a function of several variables Computer Journal, 1962, 5, 147-151

class PowerSum(dimensions=4)¶

Power sum objective function.

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

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

Where, in this exercise, $b = [8, 18, 44, 114]$

Here, $x_i \in [0, 4]$ for $i = 1, ..., 4$ .

Global optimum: $f(x) = 0$ for $x = [1, 2, 2, 3]$

Gavana, A. Global Optimization Benchmarks and AMPGO

class Price01(dimensions=2)¶

Price 1 objective function.

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

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

with $x_i \in [-500, 500]$ for $i = 1, 2$ .

Two-dimensional Price01 function

Global optimum: $f(x_i) = 0.0$ for $x = [5, 5]$ or $x = [5, -5]$ or $x = [-5, 5]$ or $x = [-5, -5]$ .

Price, W. A controlled random search procedure for global optimisation Computer Journal, 1977, 20, 367-370

class Price02(dimensions=2)¶

Price 2 objective function.

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

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

with $x_i \in [-10, 10]$ for $i = 1, 2$ .

Two-dimensional Price02 function

Global optimum: $f(x) = 0.9$ for $x_i = [0, 0]$

Price, W. A controlled random search procedure for global optimisation Computer Journal, 1977, 20, 367-370

class Price03(dimensions=2)¶

Price 3 objective function.

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

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

with $x_i \in [-50, 50]$ for $i = 1, 2$ .

Two-dimensional Price03 function

Global optimum: $f(x) = 0$ for $x = [-5, -5]$ , $x = [-5, 5]$ , $x = [5, -5]$ , $x = [5, 5]$ .

Price, W. A controlled random search procedure for global optimisation Computer Journal, 1977, 20, 367-370

Todo

Jamil #96 has an erroneous factor of 6 in front of the square brackets

class Price04(dimensions=2)¶

Price 4 objective function.

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

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

with $x_i \in [-50, 50]$ for $i = 1, 2$ .

Two-dimensional Price04 function

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

Price, W. A controlled random search procedure for global optimisation Computer Journal, 1977, 20, 367-370

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