N-D Test Functions R¶

class Rana(dimensions=2)¶

Rana objective function.

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

$f_{\text{Rana}}(x) = \sum_{i=1}^{n} \left[x_{i} \sin\left(\sqrt{\lvert{x_{1} - x_{i} + 1}\rvert}\right) \cos\left(\sqrt{\lvert{x_{1} + x_{i} + 1}\rvert}\right) + \left(x_{1} + 1\right) \sin\left(\sqrt{\lvert{x_{1} + x_{i} + 1}\rvert}\right) \cos\left(\sqrt{\lvert{x_{1} - x_{i} + 1}\rvert}\right)\right]$

Here, $n$ represents the number of dimensions and $x_i \in [-512.0, 512.0]$ for $i = 1, ..., n$ .

Two-dimensional Rana function

Global optimum: $f(x_i) = -928.5478$ for $x = [-300.3376, 500]$ .

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

homemade global minimum here.

class Rastrigin(dimensions=2)¶

Rastrigin objective function.

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

$f_{\text{Rastrigin}}(x) = 10n \sum_{i=1}^n \left[ x_i^2 - 10 \cos(2\pi x_i) \right]$

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

Two-dimensional Rastrigin function

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

Gavana, A. Global Optimization Benchmarks and AMPGO

class Ratkowsky01(dimensions=4)¶

Ratkowsky objective function.

http://www.itl.nist.gov/div898/strd/nls/data/ratkowsky3.shtml

Todo

this is a NIST regression standard dataset

class Ratkowsky02(dimensions=3)¶

Ratkowsky02 objective function.

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

$f_{\text{Ratkowsky02}}(x) = \sum_{m=1}^{9}(a_m - x[0] / (1 + exp(x[1] - b_m x[2]))^2$

where

$\begin{cases} a=[8.93, 10.8, 18.59, 22.33, 39.35, 56.11, 61.73, 64.62, 67.08] \\ b=[9., 14., 21., 28., 42., 57., 63., 70., 79.] \\ \end{cases}$

Here $x_1\in [1, 100]$ , $x_2\in [0.1, 5]$ and $x_3\in [0.01, 0.5]$

Global optimum: $f(x) = 8.0565229338$ for $x = [7.2462237576e1, 2.6180768402, 6.7359200066e-2]$

http://www.itl.nist.gov/div898/strd/nls/data/ratkowsky2.shtml

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

Two-dimensional ReduxSum function

class Ripple01(dimensions=2)¶

Ripple 1 objective function.

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

$f_{\text{Ripple01}}(x) = \sum_{i=1}^2 -e^{-2 \log 2 (\frac{x_i-0.1}{0.8})^2} \left[\sin^6(5 \pi x_i) + 0.1\cos^2(500 \pi x_i) \right]$

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

Two-dimensional Ripple01 function

Global optimum: $f(x) = -2.2$ for $x_i = 0.1$ 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 Ripple25(dimensions=2)¶

Ripple 25 objective function.

This class defines the Ripple 25 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Ripple25}}(x) = \sum_{i=1}^2 -e^{-2 \log 2 (\frac{x_i-0.1}{0.8})^2} \left[\sin^6(5 \pi x_i) \right]$

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

Two-dimensional Ripple25 function

Global optimum: $f(x) = -2$ for $x_i = 0.1$ 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 Rosenbrock(dimensions=2)¶

Rosenbrock objective function.

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

$f_{\text{Rosenbrock}}(x) = \sum_{i=1}^{n-1} [100(x_i^2 - x_{i+1})^2 + (x_i - 1)^2]$

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

Two-dimensional Rosenbrock function

Global optimum: $f(x) = 0$ for $x_i = 1$ 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 RosenbrockDisc(dimensions=2)¶: RosenbrockDisc objective function.

Two-dimensional RosenbrockDisc function

class RosenbrockModified(dimensions=2)¶

Modified Rosenbrock objective function.

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

$f_{\text{RosenbrockModified}}(x) = 74 + 100(x_2 - x_1^2)^2 + (1 - x_1)^2 - 400 e^{-\frac{(x_1+1)^2 + (x_2 + 1)^2}{0.1}}$

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

Two-dimensional RosenbrockModified function

Global optimum: $f(x) = 34.04024310$ for $x = [-0.90955374, -0.95057172]$

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

We have different global minimum compared to Jamil #106. This is possibly because of the (1-x) term is using the wrong parameter.

class RotatedEllipse01(dimensions=2)¶

Rotated Ellipse 1 objective function.

This class defines the Rotated Ellipse 1 global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{RotatedEllipse01}}(x) = 7x_1^2 - 6 \sqrt{3} x_1x_2 + 13x_2^2$

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

Two-dimensional RotatedEllipse01 function

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

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 RotatedEllipse02(dimensions=2)¶

Rotated Ellipse 2 objective function.

This class defines the Rotated Ellipse 2 global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{RotatedEllipse02}}(x) = x_1^2 - x_1 x_2 + x_2^2$

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

Two-dimensional RotatedEllipse02 function

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

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.

N-D Test Functions R¶

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