NiaPy.benchmarks

Module with implementations of benchmark functions.

class NiaPy.benchmarks.Rastrigin(Lower=-5.12, Upper=5.12)

Bases: object

Implementation of Rastrigin benchmark function.

Date: 2018

Authors: Lucija Brezočnik and Iztok Fister Jr.

License: MIT

Function: Rastrigin function

\(f(\mathbf{x}) = 10D + \sum_{i=1}^D \left(x_i^2 -10\cos(2\pi x_i)\right)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-5.12, 5.12]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = 10D + sum_{i=1}^D left(x_i^2 -10cos(2pi x_i)right)$

Equation:

begin{equation} f(mathbf{x}) = 10D + sum_{i=1}^D left(x_i^2 -10cos(2pi x_i)right) end{equation}

Domain:

$-5.12 leq x_i leq 5.12$

Reference: https://www.sfu.ca/~ssurjano/rastr.html

classmethod function()

class NiaPy.benchmarks.Rosenbrock(Lower=-30.0, Upper=30.0)

Bases: object

Implementation of Rosenbrock benchmark function.

Date: 2018

Authors: Iztok Fister Jr. and Lucija Brezočnik

License: MIT

Function: Rosenbrock function

\(f(\mathbf{x}) = \sum_{i=1}^{D-1} \left (100 (x_{i+1} - x_i^2)^2 + (x_i - 1)^2 \right)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-30, 30]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (1,...,1)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^{D-1} (100 (x_{i+1} - x_i^2)^2 + (x_i - 1)^2)$

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^{D-1} (100 (x_{i+1} - x_i^2)^2 + (x_i - 1)^2) end{equation}

Domain:

$-30 leq x_i leq 30$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Griewank(Lower=-100.0, Upper=100.0)

Bases: object

Implementation of Griewank function.

Date: 2018

Authors: Iztok Fister Jr. and Lucija Brezočnik

License: MIT

Function: Griewank function

\(f(\mathbf{x}) = \sum_{i=1}^D \frac{x_i^2}{4000} - \prod_{i=1}^D \cos(\frac{x_i}{\sqrt{i}}) + 1\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D frac{x_i^2}{4000} - prod_{i=1}^D cos(frac{x_i}{sqrt{i}}) + 1$

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^D frac{x_i^2}{4000} - prod_{i=1}^D cos(frac{x_i}{sqrt{i}}) + 1 end{equation}

Domain:

$-100 leq x_i leq 100$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Sphere(Lower=-5.12, Upper=5.12)

Bases: object

Implementation of Sphere functions.

Date: 2018

Authors: Iztok Fister Jr.

License: MIT

Function: Sphere function

\(f(\mathbf{x}) = \sum_{i=1}^D x_i^2\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [0, 10]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D x_i^2$

Equation:

begin{equation}f(mathbf{x}) = sum_{i=1}^D x_i^2 end{equation}

Domain:

$0 leq x_i leq 10$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Ackley(Lower=-32.768, Upper=32.768)

Bases: object

Implementation of Ackley function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Ackley function

\(f(\mathbf{x}) = -a\;\exp\left(-b \sqrt{\frac{1}{D}\sum_{i=1}^D x_i^2}\right) - \exp\left(\frac{1}{D}\sum_{i=1}^D \cos(c\;x_i)\right) + a + \exp(1)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-32.768, 32.768]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = -a;expleft(-b sqrt{frac{1}{D} sum_{i=1}^D x_i^2}right) - expleft(frac{1}{D} sum_{i=1}^D cos(c;x_i)right) + a + exp(1)$

Equation:

begin{equation}f(mathbf{x}) = -a;expleft(-b sqrt{frac{1}{D} sum_{i=1}^D x_i^2}right) - expleft(frac{1}{D} sum_{i=1}^D cos(c;x_i)right) + a + exp(1) end{equation}

Domain:

$-32.768 leq x_i leq 32.768$

Reference: https://www.sfu.ca/~ssurjano/ackley.html

classmethod function()

Return benchmark evaluation function.

class NiaPy.benchmarks.Schwefel(Lower=-500.0, Upper=500.0)

Bases: object

Implementation of Schewel function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Schwefel function

\(f(\textbf{x}) = 418.9829d - \sum_{i=1}^{D} x_i \sin(\sqrt{|x_i|})\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-500, 500]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (420.968746,...,420.968746)\)

LaTeX formats:

Inline:

$f(textbf{x}) = 418.9829d - sum_{i=1}^{D} x_i sin(sqrt{|x_i|})$

Equation:

begin{equation} f(textbf{x}) = 418.9829d - sum_{i=1}^{D} x_i sin(sqrt{|x_i|}) end{equation}

Domain:

$-500 leq x_i leq 500$

Reference: https://www.sfu.ca/~ssurjano/schwef.html

classmethod function()

class NiaPy.benchmarks.Schwefel221(Lower=-100.0, Upper=100.0)

Bases: object

Schwefel 2.21 function implementation.

Date: 2018

Author: Grega Vrbančič

Licence: MIT

Function: Schwefel 2.21 function

\(f(\mathbf{x})=\max_{i=1,...,D}|x_i|\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x})=max_{i=1,…,D}|x_i|$

Equation:

begin{equation}f(mathbf{x}) = max_{i=1,…,D}|x_i| end{equation}

Domain:

$-100 leq x_i leq 100$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Schwefel222(Lower=-100.0, Upper=100.0)

Bases: object

Schwefel 2.22 function implementation.

Date: 2018

Author: Grega Vrbančič

Licence: MIT

Function: Schwefel 2.22 function

\(f(\mathbf{x})=\sum_{i=1}^{D}|x_i|+\prod_{i=1}^{D}|x_i|\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x})=sum_{i=1}^{D}|x_i|+prod_{i=1}^{D}|x_i|$

Equation:

begin{equation}f(mathbf{x}) = sum_{i=1}^{D}|x_i| + prod_{i=1}^{D}|x_i| end{equation}

Domain:

$-100 leq x_i leq 100$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Whitley(Lower=-10.24, Upper=10.24)

Bases: object

Implementation of Whitley function.

Date: 2018

Authors: Grega Vrbančič and Lucija Brezočnik

License: MIT

Function: Whitley function

\(f(\mathbf{x}) = \sum_{i=1}^D \sum_{j=1}^D \left(\frac{(100(x_i^2-x_j)^2 + (1-x_j)^2)^2}{4000} - \cos(100(x_i^2-x_j)^2 + (1-x_j)^2)+1\right)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-10.24, 10.24]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (1,...,1)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D sum_{j=1}^D left(frac{(100(x_i^2-x_j)^2 + (1-x_j)^2)^2}{4000} - cos(100(x_i^2-x_j)^2 + (1-x_j)^2)+1right)$

Equation:

begin{equation}f(mathbf{x}) = sum_{i=1}^D sum_{j=1}^D left(frac{(100(x_i^2-x_j)^2 + (1-x_j)^2)^2}{4000} - cos(100(x_i^2-x_j)^2 + (1-x_j)^2)+1right) end{equation}

Domain:

$-10.24 leq x_i leq 10.24$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Alpine1(Lower=-10.0, Upper=10.0)

Bases: object

Implementation of Alpine1 function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Alpine1 function

\(f(\mathbf{x}) = \sum_{i=1}^{D} |x_i \sin(x_i)+0.1x_i|\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-10, 10]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^{D} left |x_i sin(x_i)+0.1x_i right|$

Equation:

begin{equation} f(x) = sum_{i=1}^{D} left|x_i sin(x_i) + 0.1x_i right| end{equation}

Domain:

$-10 leq x_i leq 10$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Alpine2(Lower=0.0, Upper=10.0)

Bases: object

Implementation of Alpine2 function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Alpine2 function

\(f(\mathbf{x}) = \prod_{i=1}^{D} \sqrt{x_i} \sin(x_i)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [0, 10]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 2.808^D\), at \(x^* = (7.917,...,7.917)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = prod_{i=1}^{D} sqrt{x_i} sin(x_i)$

Equation:

begin{equation} f(mathbf{x}) = prod_{i=1}^{D} sqrt{x_i} sin(x_i) end{equation}

Domain:

$0 leq x_i leq 10$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.HappyCat(Lower=-100.0, Upper=100.0)

Bases: object

Implementation of Happy cat function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Happy cat function

\(f(\mathbf{x}) = {\left |\sum_{i = 1}^D {x_i}^2 - D \right|}^{1/4} + (0.5 \sum_{i = 1}^D {x_i}^2 + \sum_{i = 1}^D x_i) / D + 0.5\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (-1,...,-1)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = {left|sum_{i = 1}^D {x_i}^2 - D right|}^{1/4} + (0.5 sum_{i = 1}^D {x_i}^2 + sum_{i = 1}^D x_i) / D + 0.5$

Equation:

begin{equation} f(mathbf{x}) = {left| sum_{i = 1}^D {x_i}^2 - D right|}^{1/4} + (0.5 sum_{i = 1}^D {x_i}^2 + sum_{i = 1}^D x_i) / D + 0.5 end{equation}

Domain:

$-100 leq x_i leq 100$

Reference: http://bee22.com/manual/tf_images/Liang%20CEC2014.pdf

classmethod function()

class NiaPy.benchmarks.Ridge(Lower=-64.0, Upper=64.0)

Bases: object

Implementation of Ridge function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Ridge function

\(f(\mathbf{x}) = \sum_{i=1}^D (\sum_{j=1}^i x_j)^2\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-64, 64]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D (sum_{j=1}^i x_j)^2 $

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^D (sum_{j=1}^i x_j)^2 end{equation}

Domain:

$-64 leq x_i leq 64$

Reference: http://www.cs.unm.edu/~neal.holts/dga/benchmarkFunction/ridge.html

classmethod function()

class NiaPy.benchmarks.ChungReynolds(Lower=-100.0, Upper=100.0)

Bases: object

Implementation of Chung Reynolds functions.

Date: 2018

Authors: Lucija Brezočnik

License: MIT

Function: Chung Reynolds function

\(f(\mathbf{x}) = \left(\sum_{i=1}^D x_i^2\right)^2\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\)

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = left(sum_{i=1}^D x_i^2right)^2$

Equation:

begin{equation} f(mathbf{x}) = left(sum_{i=1}^D x_i^2right)^2 end{equation}

Domain:

$-100 leq x_i leq 100$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Csendes(Lower=-1.0, Upper=1.0)

Bases: object

Implementation of Csendes function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Csendes function

\(f(\mathbf{x}) = \sum_{i=1}^D x_i^6\left( 2 + \sin \frac{1}{x_i}\right)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-1, 1]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D x_i^6left( 2 + sin frac{1}{x_i}right)$

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^D x_i^6left( 2 + sin frac{1}{x_i}right) end{equation}

Domain:

$-1 leq x_i leq 1$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Pinter(Lower=-10.0, Upper=10.0)

Bases: object

Implementation of Pintér function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Pintér function

\(f(\mathbf{x}) = \sum_{i=1}^D ix_i^2 + \sum_{i=1}^D 20i \sin^2 A + \sum_{i=1}^D i \log_{10} (1 + iB^2);\) \(A = (x_{i-1}\sin(x_i)+\sin(x_{i+1}))\quad \text{and} \quad\) \(B = (x_{i-1}^2 - 2x_i + 3x_{i+1} - \cos(x_i) + 1)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-10, 10]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D ix_i^2 + sum_{i=1}^D 20i sin^2 A + sum_{i=1}^D i log_{10} (1 + iB^2); A = (x_{i-1}sin(x_i)+sin(x_{i+1}))quad text{and} quad B = (x_{i-1}^2 - 2x_i + 3x_{i+1} - cos(x_i) + 1)$

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^D ix_i^2 + sum_{i=1}^D 20i sin^2 A + sum_{i=1}^D i log_{10} (1 + iB^2); A = (x_{i-1}sin(x_i)+sin(x_{i+1}))quad text{and} quad B = (x_{i-1}^2 - 2x_i + 3x_{i+1} - cos(x_i) + 1) end{equation}

Domain:

$-10 leq x_i leq 10$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Qing(Lower=-500.0, Upper=500.0)

Bases: object

Implementation of Qing function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Qing function

\(f(\mathbf{x}) = \sum_{i=1}^D \left(x_i^2 - i\right)^2\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-500, 500]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (\pm √i))\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D left (x_i^2 - iright)^2$

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^D left{(x_i^2 - iright)}^2 end{equation}

Domain:

$-500 leq x_i leq 500$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Quintic(Lower=-10.0, Upper=10.0)

Bases: object

Implementation of Quintic function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Quintic function

\(f(\mathbf{x}) = \sum_{i=1}^D \left| x_i^5 - 3x_i^4 + 4x_i^3 + 2x_i^2 - 10x_i - 4\right|\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-10, 10]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = f(-1\; \text{or}\; 2)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D left| x_i^5 - 3x_i^4 + 4x_i^3 + 2x_i^2 - 10x_i - 4right|$

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^D left| x_i^5 - 3x_i^4 + 4x_i^3 + 2x_i^2 - 10x_i - 4right| end{equation}

Domain:

$-10 leq x_i leq 10$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Salomon(Lower=-100.0, Upper=100.0)

Bases: object

Implementation of Salomon function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Salomon function

\(f(\mathbf{x}) = 1 - \cos\left(2\pi\sqrt{\sum_{i=1}^D x_i^2} \right)+ 0.1 \sqrt{\sum_{i=1}^D x_i^2}\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = f(0, 0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = 1 - cosleft(2pisqrt{sum_{i=1}^D x_i^2} right)+ 0.1 sqrt{sum_{i=1}^D x_i^2}$

Equation:

begin{equation} f(mathbf{x}) = 1 - cosleft(2pisqrt{sum_{i=1}^D x_i^2} right)+ 0.1 sqrt{sum_{i=1}^D x_i^2} end{equation}

Domain:

$-100 leq x_i leq 100$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.SchumerSteiglitz(Lower=-100.0, Upper=100.0)

Bases: object

Implementation of Schumer Steiglitz function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Schumer Steiglitz function

\(f(\mathbf{x}) = \sum_{i=1}^D x_i^4\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\).

Global minimum \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D x_i^4$

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^D x_i^4 end{equation}

Domain:

$-100 leq x_i leq 100$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Step(Lower=-100.0, Upper=100.0)

Bases: object

Implementation of Step function.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Step function

\(f(\mathbf{x}) = \sum_{i=1}^D \left( \lfloor \left | x_i \right | \rfloor \right)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D left( lfloor left | x_i right | rfloor right)$

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^D left( lfloor left | x_i right | rfloor right) end{equation}

Domain:

$-100 leq x_i leq 100$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Step2(Lower=-100.0, Upper=100.0)

Bases: object

Step2 function implementation.

Date: 2018

Author: Lucija Brezočnik

Licence: MIT

Function: Step2 function

\(f(\mathbf{x}) = \sum_{i=1}^D \left( \lfloor x_i + 0.5 \rfloor \right)^2\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\).

lobal minimum: \(f(x^*) = 0\), at \(x^* = (-0.5,...,-0.5)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D left( lfloor x_i + 0.5 rfloor right)^2$

Equation:

begin{equation}f(mathbf{x}) = sum_{i=1}^D left( lfloor x_i + 0.5 rfloor right)^2 end{equation}

Domain:

$-100 leq x_i leq 100$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Step3(Lower=-100.0, Upper=100.0)

Bases: object

Step3 function implementation.

Date: 2018

Author: Lucija Brezočnik

Licence: MIT

Function: Step3 function

\(f(\mathbf{x}) = \sum_{i=1}^D \left( \lfloor x_i^2 \rfloor \right)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-100, 100]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D left( lfloor x_i^2 rfloor right)$

Equation:

begin{equation}f(mathbf{x}) = sum_{i=1}^D left( lfloor x_i^2 rfloor right)end{equation}

Domain:

$-100 leq x_i leq 100$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.Stepint(Lower=-5.12, Upper=5.12)

Bases: object

Implementation of Stepint functions.

Date: 2018

Author: Lucija Brezočnik

License: MIT

Function: Stepint function

\(f(\mathbf{x}) = \sum_{i=1}^D x_i^2\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-5.12, 5.12]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (-5.12,...,-5.12)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D x_i^2$

Equation:

begin{equation}f(mathbf{x}) = sum_{i=1}^D x_i^2 end{equation}

Domain:

$0 leq x_i leq 10$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.SumSquares(Lower=-10.0, Upper=10.0)

Bases: object

Implementation of Sum Squares functions.

Date: 2018

Authors: Lucija Brezočnik

License: MIT

Function: Sum Squares function

\(f(\mathbf{x}) = \sum_{i=1}^D i x_i^2\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-10, 10]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = 0\), at \(x^* = (0,...,0)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = sum_{i=1}^D i x_i^2$

Equation:

begin{equation}f(mathbf{x}) = sum_{i=1}^D i x_i^2 end{equation}

Domain:

$0 leq x_i leq 10$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()

class NiaPy.benchmarks.StyblinskiTang(Lower=-5.0, Upper=5.0)

Bases: object

Implementation of Styblinski-Tang functions.

Date: 2018

Authors: Lucija Brezočnik

License: MIT

Function: Styblinski-Tang function

\(f(\mathbf{x}) = \frac{1}{2} \sum_{i=1}^D \left( x_i^4 - 16x_i^2 + 5x_i \right)\)

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube \(x_i ∈ [-5, 5]\), for all \(i = 1, 2,..., D\).

Global minimum: \(f(x^*) = -78.332\), at \(x^* = (-2.903534,...,-2.903534)\)

LaTeX formats:

Inline:

$f(mathbf{x}) = frac{1}{2} sum_{i=1}^D left( x_i^4 - 16x_i^2 + 5x_i right) $

Equation:

begin{equation}f(mathbf{x}) = frac{1}{2} sum_{i=1}^D left( x_i^4 - 16x_i^2 + 5x_i right) end{equation}

Domain:

$-5 leq x_i leq 5$

Reference paper:

Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194.

classmethod function()