# NiaPy.benchmarks¶

Module with implementations of benchmark functions.

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

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementation of Ackley function.

Date: 2018

Author: Lucija Brezočnik

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: $$f(\textbf{x}^*) = 0$$, 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

Initialize of Ackley benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['Ackley']
__init__(Lower=- 32.768, Upper=32.768)[source]

Initialize of Ackley benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

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

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementation of Alpine1 function.

Date: 2018

Author: Lucija Brezočnik

Function: Alpine1 function

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

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:

$f(mathbf{x}) = sum_{i=1}^{D} lvert x_i sin(x_i)+0.1x_i rvert$

Equation:

begin{equation} f(mathbf{x}) = sum_{i=1}^{D} lvert x_i sin(x_i)+0.1x_i rvert 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.

Initialize of Alpine1 benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['Alpine1']
__init__(Lower=- 10.0, Upper=10.0)[source]

Initialize of Alpine1 benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

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

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementation of Alpine2 function.

Date: 2018

Author: Lucija Brezočnik

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.

Initialize of Alpine2 benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['Alpine2']
__init__(Lower=0.0, Upper=10.0)[source]

Initialize of Alpine2 benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

class NiaPy.benchmarks.Benchmark(Lower, Upper, **kwargs)[source]

Bases: object

Class representing benchmarks.

Variables
• Name (List[str]) – List of names representiong benchmark names.

• Lower (Union[int, float, list, numpy.ndarray]) – Lower bounds.

• Upper (Union[int, float, list, numpy.ndarray]) – Upper bounds.

Initialize benchmark.

Parameters
• Lower (Union[int, float, list, numpy.ndarray]) – Lower bounds.

• Upper (Union[int, float, list, numpy.ndarray]) – Upper bounds.

• **kwargs (Dict[str, Any]) – Additional arguments.

Name = ['Benchmark', 'BBB']
__call__()[source]

Get the optimization function.

Returns

Fitness funciton.

Return type

Callable[[int, Union[list, numpy.ndarray]], float]

__init__(Lower, Upper, **kwargs)[source]

Initialize benchmark.

Parameters
• Lower (Union[int, float, list, numpy.ndarray]) – Lower bounds.

• Upper (Union[int, float, list, numpy.ndarray]) – Upper bounds.

• **kwargs (Dict[str, Any]) – Additional arguments.

function()[source]

Get the optimization function.

Returns

Fitness funciton.

Return type

Callable[[int, Union[list, numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

plot2d()[source]

Plot 2D graph.

plot3d(scale=0.32)[source]

Plot 3d scatter plot of benchmark function.

Parameters

scale (float) – Scale factor for points.

class NiaPy.benchmarks.BentCigar(Lower=- 100.0, Upper=100.0)[source]

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementations of Bent Cigar functions.

Date: 2018

Author: Klemen Berkovič

Function: Bent Cigar Function

$$f(\textbf{x}) = x_1^2 + 10^6 \sum_{i=2}^D x_i^2$$

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 $$x^* = (420.968746,...,420.968746)$$

LaTeX formats:
Inline:

$f(textbf{x}) = x_1^2 + 10^6 sum_{i=2}^D x_i^2$

Equation:

begin{equation} f(textbf{x}) = x_1^2 + 10^6 sum_{i=2}^D x_i^2 end{equation}

Domain:

$-100 leq x_i leq 100$

Reference:

Initialize of Bent Cigar benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['BentCigar']
__init__(Lower=- 100.0, Upper=100.0)[source]

Initialize of Bent Cigar benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

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

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementation of Chung Reynolds functions.

Date: 2018

Authors: Lucija Brezočnik

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.

Initialize of Chung Reynolds benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['ChungReynolds']
__init__(Lower=- 100.0, Upper=100.0)[source]

Initialize of Chung Reynolds benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

class NiaPy.benchmarks.CosineMixture(Lower=- 1.0, Upper=1.0)[source]

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementations of Cosine mixture function.

Date: 2018

Author: Klemen Berkovič

Function: Cosine Mixture Function

$$f(\textbf{x}) = - 0.1 \sum_{i = 1}^D \cos (5 \pi x_i) - \sum_{i = 1}^D x_i^2$$

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

Global maximu: $$f(x^*) = -0.1 D$$, at $$x^* = (0.0,...,0.0)$$

LaTeX formats:
Inline:

$f(textbf{x}) = - 0.1 sum_{i = 1}^D cos (5 pi x_i) - sum_{i = 1}^D x_i^2$

Equation:

begin{equation} f(textbf{x}) = - 0.1 sum_{i = 1}^D cos (5 pi x_i) - sum_{i = 1}^D x_i^2 end{equation}

Domain:

$-1 leq x_i leq 1$

Initialize of Cosine mixture benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['CosineMixture']
__init__(Lower=- 1.0, Upper=1.0)[source]

Initialize of Cosine mixture benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

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

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementation of Csendes function.

Date: 2018

Author: Lucija Brezočnik

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.

Initialize of Csendes benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['Csendes']
__init__(Lower=- 1.0, Upper=1.0)[source]

Initialize of Csendes benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

class NiaPy.benchmarks.Discus(Lower=- 100.0, Upper=100.0)[source]

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementations of Discus functions.

Date: 2018

Author: Klemen Berkovič

Function: Discus Function

$$f(\textbf{x}) = x_1^2 10^6 + \sum_{i=2}^D x_i^2$$

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 $$x^* = (420.968746,...,420.968746)$$

LaTeX formats:
Inline:

$f(textbf{x}) = x_1^2 10^6 + sum_{i=2}^D x_i^2$

Equation:

begin{equation} f(textbf{x}) = x_1^2 10^6 + sum_{i=2}^D x_i^2 end{equation}

Domain:

$-100 leq x_i leq 100$

Initialize of Discus benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['Discus']
__init__(Lower=- 100.0, Upper=100.0)[source]

Initialize of Discus benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

class NiaPy.benchmarks.DixonPrice(Lower=- 10.0, Upper=10)[source]

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementations of Dixon Price function.

Date: 2018

Author: Klemen Berkovič

Function: Levy Function

$$f(\textbf{x}) = (x_1 - 1)^2 + \sum_{i = 2}^D i (2x_i^2 - x_{i - 1})^2$$

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: $$f(\textbf{x}^*) = 0$$ at $$\textbf{x}^* = (2^{-\frac{2^1 - 2}{2^1}}, \cdots , 2^{-\frac{2^i - 2}{2^i}} , \cdots , 2^{-\frac{2^D - 2}{2^D}})$$

LaTeX formats:
Inline:

$f(textbf{x}) = (x_1 - 1)^2 + sum_{i = 2}^D i (2x_i^2 - x_{i - 1})^2$

Equation:

begin{equation} f(textbf{x}) = (x_1 - 1)^2 + sum_{i = 2}^D i (2x_i^2 - x_{i - 1})^2 end{equation}

Domain:

$-10 leq x_i leq 10$

Initialize of Dixon Price benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['DixonPrice']
__init__(Lower=- 10.0, Upper=10)[source]

Initialize of Dixon Price benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

class NiaPy.benchmarks.Elliptic(Lower=- 100.0, Upper=100.0)[source]

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementations of High Conditioned Elliptic functions.

Date: 2018

Author: Klemen Berkovič

Function: High Conditioned Elliptic Function

$$f(\textbf{x}) = \sum_{i=1}^D \left( 10^6 \right)^{ \frac{i - 1}{D - 1} } x_i^2$$

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 $$x^* = (420.968746,...,420.968746)$$

LaTeX formats:
Inline:

$f(textbf{x}) = sum_{i=1}^D left( 10^6 right)^{ frac{i - 1}{D - 1} } x_i^2$

Equation:

begin{equation} f(textbf{x}) = sum_{i=1}^D left( 10^6 right)^{ frac{i - 1}{D - 1} } x_i^2 end{equation}

Domain:

$-100 leq x_i leq 100$

Initialize of High Conditioned Elliptic benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['Elliptic']
__init__(Lower=- 100.0, Upper=100.0)[source]

Initialize of High Conditioned Elliptic benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

class NiaPy.benchmarks.ExpandedGriewankPlusRosenbrock(Lower=- 100.0, Upper=100.0)[source]

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementation of Expanded Griewank’s plus Rosenbrock function.

Date: 2018

Author: Klemen Berkovič

Function: Expanded Griewank’s plus Rosenbrock function

$$f(\textbf{x}) = h(g(x_D, x_1)) + \sum_{i=2}^D h(g(x_{i - 1}, x_i)) \\ g(x, y) = 100 (x^2 - y)^2 + (x - 1)^2 \\ h(z) = \frac{z^2}{4000} - \cos \left( \frac{z}{\sqrt{1}} \right) + 1$$

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 $$x^* = (420.968746,...,420.968746)$$

LaTeX formats:
Inline:

$f(textbf{x}) = h(g(x_D, x_1)) + sum_{i=2}^D h(g(x_{i - 1}, x_i)) \ g(x, y) = 100 (x^2 - y)^2 + (x - 1)^2 \ h(z) = frac{z^2}{4000} - cos left( frac{z}{sqrt{1}} right) + 1$

Equation:

begin{equation} f(textbf{x}) = h(g(x_D, x_1)) + sum_{i=2}^D h(g(x_{i - 1}, x_i)) \ g(x, y) = 100 (x^2 - y)^2 + (x - 1)^2 \ h(z) = frac{z^2}{4000} - cos left( frac{z}{sqrt{1}} right) + 1 end{equation}

Domain:

$-100 leq x_i leq 100$

Reference:

Initialize of Expanded Griewank’s plus Rosenbrock benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['ExpandedGriewankPlusRosenbrock']
__init__(Lower=- 100.0, Upper=100.0)[source]

Initialize of Expanded Griewank’s plus Rosenbrock benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

class NiaPy.benchmarks.ExpandedSchaffer(Lower=- 100.0, Upper=100.0)[source]

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementations of Expanded Schaffer functions.

Date: 2018

Author: Klemen Berkovič

Function: Expanded Schaffer Function $$f(\textbf{x}) = g(x_D, x_1) + \sum_{i=2}^D g(x_{i - 1}, x_i) \\ g(x, y) = 0.5 + \frac{\sin \left(\sqrt{x^2 + y^2} \right)^2 - 0.5}{\left( 1 + 0.001 (x^2 + y^2) \right)}^2$$

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 $$x^* = (420.968746,...,420.968746)$$

LaTeX formats: Inline: $f(textbf{x}) = g(x_D, x_1) + sum_{i=2}^D g(x_{i - 1}, x_i) \ g(x, y) = 0.5 + frac{sin left(sqrt{x^2 + y^2} right)^2 - 0.5}{left( 1 + 0.001 (x^2 + y^2) right)}^2$

Equation: begin{equation} f(textbf{x}) = g(x_D, x_1) + sum_{i=2}^D g(x_{i - 1}, x_i) \ g(x, y) = 0.5 + frac{sin left(sqrt{x^2 + y^2} right)^2 - 0.5}{left( 1 + 0.001 (x^2 + y^2) right)}^2 end{equation}

Domain: $-100 leq x_i leq 100$

Reference:

Initialize of Expanded Scaffer benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['ExpandedSchaffer']
__init__(Lower=- 100.0, Upper=100.0)[source]

Initialize of Expanded Scaffer benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

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

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementation of Griewank function.

Date: 2018

Authors: Iztok Fister Jr. and Lucija Brezočnik

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.

Initialize of Griewank benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['Griewank']
__init__(Lower=- 100.0, Upper=100.0)[source]

Initialize of Griewank benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

class NiaPy.benchmarks.HGBat(Lower=- 100.0, Upper=100.0)[source]

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementations of HGBat functions.

Date: 2018

Author: Klemen Berkovič

Function:

HGBat Function $$f(\textbf{x}) = \left| \left( \sum_{i=1}^D x_i^2 \right)^2 - \left( \sum_{i=1}^D x_i \right)^2 \right|^{\frac{1}{2}} + \frac{0.5 \sum_{i=1}^D x_i^2 + \sum_{i=1}^D x_i}{D} + 0.5$$

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 $$x^* = (420.968746,...,420.968746)$$

LaTeX formats:
Inline:

$$f(textbf{x}) = left| left( sum_{i=1}^D x_i^2 right)^2 - left( sum_{i=1}^D x_i right)^2 right|^{frac{1}{2}} + frac{0.5 sum_{i=1}^D x_i^2 + sum_{i=1}^D x_i}{D} + 0.5 Equation: begin{equation} f(textbf{x}) = left| left( sum_{i=1}^D x_i^2 right)^2 - left( sum_{i=1}^D x_i right)^2 right|^{frac{1}{2}} + frac{0.5 sum_{i=1}^D x_i^2 + sum_{i=1}^D x_i}{D} + 0.5 end{equation} Domain: -100 leq x_i leq 100 Reference: http://www5.zzu.edu.cn/__local/A/69/BC/D3B5DFE94CD2574B38AD7CD1D12_C802DAFE_BC0C0.pdf Initialize of HGBat benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['HGBat'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of HGBat benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.HappyCat(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Happy cat function. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Happy cat function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = {left|sum_{i = 1}^D {x_i}^2 - D right|}^{1/4} + (0.5 sum_{i = 1}^D {x_i}^2 + sum_{i = 1}^D x_i) / D + 0.5 Equation: begin{equation} f(mathbf{x}) = {left| sum_{i = 1}^D {x_i}^2 - D right|}^{1/4} + (0.5 sum_{i = 1}^D {x_i}^2 + sum_{i = 1}^D x_i) / D + 0.5 end{equation} Domain: -100 leq x_i leq 100 Reference: http://bee22.com/manual/tf_images/Liang%20CEC2014.pdf & Beyer, H. G., & Finck, S. (2012). HappyCat - A Simple Function Class Where Well-Known Direct Search Algorithms Do Fail. In International Conference on Parallel Problem Solving from Nature (pp. 367-376). Springer, Berlin, Heidelberg. Initialize of Happy cat benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['HappyCat'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of Happy cat benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Infinity(Lower=- 1.0, Upper=1.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Infinity function. Date: 2018 Author: Klemen Berkovič License: MIT Function: Infinity Function $$f(\textbf{x}) = \sum_{i = 1}^D x_i^6 \left( \sin \left( \frac{1}{x_i} \right) + 2 \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^* = (420.968746,...,420.968746)$$ LaTeX formats: Inline: f(textbf{x}) = sum_{i = 1}^D x_i^6 left( sin left( frac{1}{x_i} right) + 2 right) Equation: begin{equation} f(textbf{x}) = sum_{i = 1}^D x_i^6 left( sin left( frac{1}{x_i} right) + 2 right) end{equation} Domain: -1 leq x_i leq 1 Reference: http://infinity77.net/global_optimization/test_functions_nd_I.html#go_benchmark.Infinity Initialize of Infinity benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Infinity'] __init__(Lower=- 1.0, Upper=1.0)[source] Initialize of Infinity benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Katsuura(Lower=- 100.0, Upper=100.0, **kwargs)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Katsuura functions. Date: 2018 Author: Klemen Berkovič License: MIT Function: Katsuura Function $$f(\textbf{x}) = \frac{10}{D^2} \prod_{i=1}^D \left( 1 + i \sum_{j=1}^{32} \frac{\lvert 2^j x_i - round\left(2^j x_i \right) \rvert}{2^j} \right)^\frac{10}{D^{1.2}} - \frac{10}{D^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^* = (420.968746,...,420.968746)$$ LaTeX formats: Inline: f(textbf{x}) = frac{10}{D^2} prod_{i=1}^D left( 1 + i sum_{j=1}^{32} frac{lvert 2^j x_i - roundleft(2^j x_i right) rvert}{2^j} right)^frac{10}{D^{1.2}} - frac{10}{D^2} Equation: begin{equation} f(textbf{x}) = frac{10}{D^2} prod_{i=1}^D left( 1 + i sum_{j=1}^{32} frac{lvert 2^j x_i - roundleft(2^j x_i right) rvert}{2^j} right)^frac{10}{D^{1.2}} - frac{10}{D^2} end{equation} Domain: -100 leq x_i leq 100 Reference: http://www5.zzu.edu.cn/__local/A/69/BC/D3B5DFE94CD2574B38AD7CD1D12_C802DAFE_BC0C0.pdf Initialize of Katsuura benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Katsuura'] __init__(Lower=- 100.0, Upper=100.0, **kwargs)[source] Initialize of Katsuura benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Levy(Lower=0.0, Upper=3.141592653589793)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Levy functions. Date: 2018 Author: Klemen Berkovič License: MIT Function: Levy Function $$f(\textbf{x}) = \sin^2 (\pi w_1) + \sum_{i = 1}^{D - 1} (w_i - 1)^2 \left( 1 + 10 \sin^2 (\pi w_i + 1) \right) + (w_d - 1)^2 (1 + \sin^2 (2 \pi w_d)) \\ w_i = 1 + \frac{x_i - 1}{4}$$ Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube $$x_i ∈ [-10, 10]$$, for all $$i = 1, 2,..., D$$. Global minimum: $$f(\textbf{x}^*) = 0$$ at $$\textbf{x}^* = (1, \cdots, 1)$$ LaTeX formats: Inline: f(textbf{x}) = sin^2 (pi w_1) + sum_{i = 1}^{D - 1} (w_i - 1)^2 left( 1 + 10 sin^2 (pi w_i + 1) right) + (w_d - 1)^2 (1 + sin^2 (2 pi w_d)) \ w_i = 1 + frac{x_i - 1}{4} Equation: begin{equation} f(textbf{x}) = sin^2 (pi w_1) + sum_{i = 1}^{D - 1} (w_i - 1)^2 left( 1 + 10 sin^2 (pi w_i + 1) right) + (w_d - 1)^2 (1 + sin^2 (2 pi w_d)) \ w_i = 1 + frac{x_i - 1}{4} end{equation} Domain: -10 leq x_i leq 10 Reference: https://www.sfu.ca/~ssurjano/levy.html Initialize of Levy benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Levy'] __init__(Lower=0.0, Upper=3.141592653589793)[source] Initialize of Levy benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Michalewichz(Lower=0.0, Upper=3.141592653589793, m=10)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Michalewichz’s functions. Date: 2018 Author: Klemen Berkovič License: MIT Function: High Conditioned Elliptic Function $$f(\textbf{x}) = \sum_{i=1}^D \left( 10^6 \right)^{ \frac{i - 1}{D - 1} } 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, \pi]$$, for all $$i = 1, 2,..., D$$. Global minimum: at $$d = 2$$ $$f(\textbf{x}^*) = -1.8013$$ at $$\textbf{x}^* = (2.20, 1.57)$$ at $$d = 5$$ $$f(\textbf{x}^*) = -4.687658$$ at $$d = 10$$ $$f(\textbf{x}^*) = -9.66015$$ LaTeX formats: Inline: f(textbf{x}) = - sum_{i = 1}^{D} sin(x_i) sinleft( frac{ix_i^2}{pi} right)^{2m} Equation: begin{equation} f(textbf{x}) = - sum_{i = 1}^{D} sin(x_i) sinleft( frac{ix_i^2}{pi} right)^{2m} end{equation} Domain: 0 leq x_i leq pi Reference URL: https://www.sfu.ca/~ssurjano/michal.html Initialize of Michalewichz benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Michalewichz'] __init__(Lower=0.0, Upper=3.141592653589793, m=10)[source] Initialize of Michalewichz benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. classmethod function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.ModifiedSchwefel(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Modified Schwefel functions. Date: 2018 Author: Klemen Berkovič License: MIT Function: Modified Schwefel Function $$f(\textbf{x}) = 418.9829 \cdot D - \sum_{i=1}^D h(x_i) \\ h(x) = g(x + 420.9687462275036) \\ g(z) = \begin{cases} z \sin \left( \lvert z \rvert^{\frac{1}{2}} \right) &\quad \lvert z \rvert \leq 500 \\ \left( 500 - \mod (z, 500) \right) \sin \left( \sqrt{\lvert 500 - \mod (z, 500) \rvert} \right) - \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z > 500 \\ \left( \mod (\lvert z \rvert, 500) - 500 \right) \sin \left( \sqrt{\lvert \mod (\lvert z \rvert, 500) - 500 \rvert} \right) + \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z < -500\end{cases}$$ Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube $$x_i ∈ [-100, 100]$$, for all $$i = 1, 2,..., D$$. Global minimum: $$f(x^*) = 0$$, at $$x^* = (420.968746,...,420.968746)$$ LaTeX formats: Inline: f(textbf{x}) = 418.9829 cdot D - sum_{i=1}^D h(x_i) \ h(x) = g(x + 420.9687462275036) \ g(z) = begin{cases} z sin left( lvert z rvert^{frac{1}{2}} right) &quad lvert z rvert leq 500 \ left( 500 - mod (z, 500) right) sin left( sqrt{lvert 500 - mod (z, 500) rvert} right) - frac{ left( z - 500 right)^2 }{ 10000 D } &quad z > 500 \ left( mod (lvert z rvert, 500) - 500 right) sin left( sqrt{lvert mod (lvert z rvert, 500) - 500 rvert} right) + frac{ left( z - 500 right)^2 }{ 10000 D } &quad z < -500end{cases} Equation: begin{equation} f(textbf{x}) = 418.9829 cdot D - sum_{i=1}^D h(x_i) \ h(x) = g(x + 420.9687462275036) \ g(z) = begin{cases} z sin left( lvert z rvert^{frac{1}{2}} right) &quad lvert z rvert leq 500 \ left( 500 - mod (z, 500) right) sin left( sqrt{lvert 500 - mod (z, 500) rvert} right) - frac{ left( z - 500 right)^2 }{ 10000 D } &quad z > 500 \ left( mod (lvert z rvert, 500) - 500 right) sin left( sqrt{lvert mod (lvert z rvert, 500) - 500 rvert} right) + frac{ left( z - 500 right)^2 }{ 10000 D } &quad z < -500end{cases} end{equation} Domain: -100 leq x_i leq 100 Reference: http://www5.zzu.edu.cn/__local/A/69/BC/D3B5DFE94CD2574B38AD7CD1D12_C802DAFE_BC0C0.pdf Initialize of Modified Schwefel benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['ModifiedSchwefel'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of Modified Schwefel benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Perm(D=10.0, beta=0.5)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Perm functions. Date: 2018 Author: Klemen Berkovič License: MIT Arguments: beta {real} – value added to inner sum of funciton Function: Perm Function $$f(\textbf{x}) = \sum_{i = 1}^D \left( \sum_{j = 1}^D (j - \beta) \left( x_j^i - \frac{1}{j^i} \right) \right)^2$$ Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube $$x_i ∈ [-D, D]$$, for all $$i = 1, 2,..., D$$. Global minimum: $$f(\textbf{x}^*) = 0$$ at $$\textbf{x}^* = (1, \frac{1}{2}, \cdots , \frac{1}{i} , \cdots , \frac{1}{D})$$ LaTeX formats: Inline: f(textbf{x}) = sum_{i = 1}^D left( sum_{j = 1}^D (j - beta) left( x_j^i - frac{1}{j^i} right) right)^2 Equation: begin{equation} f(textbf{x}) = sum_{i = 1}^D left( sum_{j = 1}^D (j - beta) left( x_j^i - frac{1}{j^i} right) right)^2 end{equation} Domain: -D leq x_i leq D Reference: https://www.sfu.ca/~ssurjano/perm0db.html Initialize of Bent Cigar benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Perm'] __init__(D=10.0, beta=0.5)[source] Initialize of Bent Cigar benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Pinter(Lower=- 10.0, Upper=10.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Pintér function. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Pintér function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D ix_i^2 + sum_{i=1}^D 20i sin^2 A + sum_{i=1}^D i log_{10} (1 + iB^2); A = (x_{i-1}sin(x_i)+sin(x_{i+1}))quad text{and} quad B = (x_{i-1}^2 - 2x_i + 3x_{i+1} - cos(x_i) + 1) Equation: begin{equation} f(mathbf{x}) = sum_{i=1}^D ix_i^2 + sum_{i=1}^D 20i sin^2 A + sum_{i=1}^D i log_{10} (1 + iB^2); A = (x_{i-1}sin(x_i)+sin(x_{i+1}))quad text{and} quad B = (x_{i-1}^2 - 2x_i + 3x_{i+1} - cos(x_i) + 1) end{equation} Domain: -10 leq x_i leq 10 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Pinter benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Pinter'] __init__(Lower=- 10.0, Upper=10.0)[source] Initialize of Pinter benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Powell(Lower=- 4.0, Upper=5.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Powell functions. Date: 2018 Author: Klemen Berkovič License: MIT Function: Levy Function $$f(\textbf{x}) = \sum_{i = 1}^{D / 4} \left( (x_{4 i - 3} + 10 x_{4 i - 2})^2 + 5 (x_{4 i - 1} - x_{4 i})^2 + (x_{4 i - 2} - 2 x_{4 i - 1})^4 + 10 (x_{4 i - 3} - x_{4 i})^4 \right)$$ Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube $$x_i ∈ [-4, 5]$$, for all $$i = 1, 2,..., D$$. Global minimum: $$f(\textbf{x}^*) = 0$$ at $$\textbf{x}^* = (0, \cdots, 0)$$ LaTeX formats: Inline: f(textbf{x}) = sum_{i = 1}^{D / 4} left( (x_{4 i - 3} + 10 x_{4 i - 2})^2 + 5 (x_{4 i - 1} - x_{4 i})^2 + (x_{4 i - 2} - 2 x_{4 i - 1})^4 + 10 (x_{4 i - 3} - x_{4 i})^4 right) Equation: begin{equation} f(textbf{x}) = sum_{i = 1}^{D / 4} left( (x_{4 i - 3} + 10 x_{4 i - 2})^2 + 5 (x_{4 i - 1} - x_{4 i})^2 + (x_{4 i - 2} - 2 x_{4 i - 1})^4 + 10 (x_{4 i - 3} - x_{4 i})^4 right) end{equation} Domain: -4 leq x_i leq 5 Reference: https://www.sfu.ca/~ssurjano/levy.html Initialize of Powell benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Powell'] __init__(Lower=- 4.0, Upper=5.0)[source] Initialize of Powell benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Qing(Lower=- 500.0, Upper=500.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Qing function. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Qing function $$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))$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D left (x_i^2 - iright)^2 Equation: begin{equation} f(mathbf{x}) = sum_{i=1}^D left{(x_i^2 - iright)}^2 end{equation} Domain: -500 leq x_i leq 500 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Qing benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Qing'] __init__(Lower=- 500.0, Upper=500.0)[source] Initialize of Qing benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Quintic(Lower=- 10.0, Upper=10.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Quintic function. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Quintic function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D left| x_i^5 - 3x_i^4 + 4x_i^3 + 2x_i^2 - 10x_i - 4right| Equation: begin{equation} f(mathbf{x}) = sum_{i=1}^D left| x_i^5 - 3x_i^4 + 4x_i^3 + 2x_i^2 - 10x_i - 4right| end{equation} Domain: -10 leq x_i leq 10 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Quintic benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Quintic'] __init__(Lower=- 10.0, Upper=10.0)[source] Initialize of Quintic benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Rastrigin(Lower=- 5.12, Upper=5.12)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Rastrigin benchmark function. Date: 2018 Authors: Lucija Brezočnik and Iztok Fister Jr. License: MIT Function: Rastrigin function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = 10D + sum_{i=1}^D left(x_i^2 -10cos(2pi x_i)right) Equation: begin{equation} f(mathbf{x}) = 10D + sum_{i=1}^D left(x_i^2 -10cos(2pi x_i)right) end{equation} Domain: -5.12 leq x_i leq 5.12 Reference: https://www.sfu.ca/~ssurjano/rastr.html Initialize of Rastrigni benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Rastrigin'] __init__(Lower=- 5.12, Upper=5.12)[source] Initialize of Rastrigni benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Ridge(Lower=- 64.0, Upper=64.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Ridge function. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Ridge function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D (sum_{j=1}^i x_j)^2  Equation: begin{equation} f(mathbf{x}) = sum_{i=1}^D (sum_{j=1}^i x_j)^2 end{equation} Domain: -64 leq x_i leq 64 Initialize of Ridge benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Ridge'] __init__(Lower=- 64.0, Upper=64.0)[source] Initialize of Ridge benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Rosenbrock(Lower=- 30.0, Upper=30.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Rosenbrock benchmark function. Date: 2018 Authors: Iztok Fister Jr. and Lucija Brezočnik License: MIT Function: Rosenbrock function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^{D-1} (100 (x_{i+1} - x_i^2)^2 + (x_i - 1)^2) Equation: begin{equation} f(mathbf{x}) = sum_{i=1}^{D-1} (100 (x_{i+1} - x_i^2)^2 + (x_i - 1)^2) end{equation} Domain: -30 leq x_i leq 30 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Rosenbrock benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Rosenbrock'] __init__(Lower=- 30.0, Upper=30.0)[source] Initialize of Rosenbrock benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Salomon(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Salomon function. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Salomon function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = 1 - cosleft(2pisqrt{sum_{i=1}^D x_i^2} right)+ 0.1 sqrt{sum_{i=1}^D x_i^2} Equation: begin{equation} f(mathbf{x}) = 1 - cosleft(2pisqrt{sum_{i=1}^D x_i^2} right)+ 0.1 sqrt{sum_{i=1}^D x_i^2} end{equation} Domain: -100 leq x_i leq 100 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Salomon benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Salomon'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of Salomon benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.SchafferN2(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Schaffer N. 2 functions. Date: 2018 Author: Klemen Berkovič License: MIT Function: Schaffer N. 2 Function $$f(\textbf{x}) = 0.5 + \frac{ \sin^2 \left( x_1^2 - x_2^2 \right) - 0.5 }{ \left( 1 + 0.001 \left( x_1^2 + x_2^2 \right) \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^* = (420.968746,...,420.968746)$$ LaTeX formats: Inline: f(textbf{x}) = 0.5 + frac{ sin^2 left( x_1^2 - x_2^2 right) - 0.5 }{ left( 1 + 0.001 left( x_1^2 + x_2^2 right) right)^2 } Equation: begin{equation} f(textbf{x}) = 0.5 + frac{ sin^2 left( x_1^2 - x_2^2 right) - 0.5 }{ left( 1 + 0.001 left( x_1^2 + x_2^2 right) right)^2 } end{equation} Domain: -100 leq x_i leq 100 Reference: http://www5.zzu.edu.cn/__local/A/69/BC/D3B5DFE94CD2574B38AD7CD1D12_C802DAFE_BC0C0.pdf Initialize of SchafferN2 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['SchafferN2'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of SchafferN2 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.SchafferN4(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Schaffer N. 2 functions. Date: 2018 Author: Klemen Berkovič License: MIT Function: Schaffer N. 2 Function $$f(\textbf{x}) = 0.5 + \frac{ \cos^2 \left( \sin \left( x_1^2 - x_2^2 \right) \right)- 0.5 }{ \left( 1 + 0.001 \left( x_1^2 + x_2^2 \right) \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^* = (420.968746,...,420.968746)$$ LaTeX formats: Inline: f(textbf{x}) = 0.5 + frac{ cos^2 left( sin left( x_1^2 - x_2^2 right) right)- 0.5 }{ left( 1 + 0.001 left( x_1^2 + x_2^2 right) right)^2 } Equation: begin{equation} f(textbf{x}) = 0.5 + frac{ cos^2 left( sin left( x_1^2 - x_2^2 right) right)- 0.5 }{ left( 1 + 0.001 left( x_1^2 + x_2^2 right) right)^2 } end{equation} Domain: -100 leq x_i leq 100 Reference: http://www5.zzu.edu.cn/__local/A/69/BC/D3B5DFE94CD2574B38AD7CD1D12_C802DAFE_BC0C0.pdf Initialize of ScahfferN4 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['SchafferN4'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of ScahfferN4 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.SchumerSteiglitz(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Schumer Steiglitz function. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Schumer Steiglitz function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D x_i^4 Equation: begin{equation} f(mathbf{x}) = sum_{i=1}^D x_i^4 end{equation} Domain: -100 leq x_i leq 100 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Schumer Steiglitz benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['SchumerSteiglitz'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of Schumer Steiglitz benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Schwefel(Lower=- 500.0, Upper=500.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Schewel function. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Schwefel function $$f(\textbf{x}) = 418.9829d - \sum_{i=1}^{D} x_i \sin(\sqrt{\lvert x_i \rvert})$$ Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube $$x_i ∈ [-500, 500]$$, for all $$i = 1, 2,..., D$$. Global minimum: $$f(x^*) = 0$$, at $$x^* = (420.968746,...,420.968746)$$ LaTeX formats: Inline: f(textbf{x}) = 418.9829d - sum_{i=1}^{D} x_i sin(sqrt{lvert x_i rvert}) Equation: begin{equation} f(textbf{x}) = 418.9829d - sum_{i=1}^{D} x_i sin(sqrt{lvert x_i rvert}) end{equation} Domain: -500 leq x_i leq 500 Reference: https://www.sfu.ca/~ssurjano/schwef.html Initialize of Schwefel benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Schwefel'] __init__(Lower=- 500.0, Upper=500.0)[source] Initialize of Schwefel benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Schwefel221(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Schwefel 2.21 function implementation. Date: 2018 Author: Grega Vrbančič Licence: MIT Function: Schwefel 2.21 function $$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)$$ LaTeX formats: Inline: f(mathbf{x})=max_{i=1,…,D} lvert x_i rvert Equation: begin{equation}f(mathbf{x}) = max_{i=1,…,D} lvert x_i rvert end{equation} Domain: -100 leq x_i leq 100 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Schwefel221 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Schwefel221'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of Schwefel221 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Schwefel222(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Schwefel 2.22 function implementation. Date: 2018 Author: Grega Vrbančič Licence: MIT Function: Schwefel 2.22 function $$f(\mathbf{x})=\sum_{i=1}^{D} \lvert x_i \rvert +\prod_{i=1}^{D} \lvert x_i \rvert$$ Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube $$x_i ∈ [-100, 100]$$, for all $$i = 1, 2,..., D$$. Global minimum: $$f(x^*) = 0$$, at $$x^* = (0,...,0)$$ LaTeX formats: Inline: f(mathbf{x})=sum_{i=1}^{D} lvert x_i rvert +prod_{i=1}^{D} lvert x_i rvert Equation: begin{equation}f(mathbf{x}) = sum_{i=1}^{D} lvert x_i rvert + prod_{i=1}^{D} lvert x_i rvert end{equation} Domain: -100 leq x_i leq 100 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Schwefel222 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Schwefel222'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of Schwefel222 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Sphere(Lower=- 5.12, Upper=5.12)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Sphere functions. Date: 2018 Authors: Iztok Fister Jr. License: MIT Function: Sphere function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D x_i^2 Equation: begin{equation}f(mathbf{x}) = sum_{i=1}^D x_i^2 end{equation} Domain: 0 leq x_i leq 10 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Sphere benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Sphere'] __init__(Lower=- 5.12, Upper=5.12)[source] Initialize of Sphere benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Sphere2(Lower=- 1.0, Upper=1.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Sphere with different powers function. Date: 2018 Authors: Klemen Berkovič License: MIT Function: Sun of different powers function $$f(\textbf{x}) = \sum_{i = 1}^D \lvert x_i \rvert^{i + 1}$$ Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube $$x_i ∈ [-1, 1]$$, for all $$i = 1, 2,..., D$$. Global minimum: $$f(x^*) = 0$$, at $$x^* = (0,...,0)$$ LaTeX formats: Inline: f(textbf{x}) = sum_{i = 1}^D lvert x_i rvert^{i + 1} Equation: begin{equation} f(textbf{x}) = sum_{i = 1}^D lvert x_i rvert^{i + 1} end{equation} Domain: -1 leq x_i leq 1 Reference URL: https://www.sfu.ca/~ssurjano/sumpow.html Initialize of Sphere2 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Sphere2'] __init__(Lower=- 1.0, Upper=1.0)[source] Initialize of Sphere2 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Sphere3(Lower=- 65.536, Upper=65.536)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of rotated hyper-ellipsoid function. Date: 2018 Authors: Klemen Berkovič License: MIT Function: Sun of rotated hyper-elliposid function $$f(\textbf{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 ∈ [-65.536, 65.536]$$, for all $$i = 1, 2,..., D$$. Global minimum: $$f(x^*) = 0$$, at $$x^* = (0,...,0)$$ LaTeX formats: Inline: f(textbf{x}) = sum_{i = 1}^D sum_{j = 1}^i x_j^2 Equation: begin{equation} f(textbf{x}) = sum_{i = 1}^D sum_{j = 1}^i x_j^2 end{equation} Domain: -65.536 leq x_i leq 65.536 Reference URL: https://www.sfu.ca/~ssurjano/rothyp.html Initialize of Sphere3 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Sphere3'] __init__(Lower=- 65.536, Upper=65.536)[source] Initialize of Sphere3 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Step(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Step function. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Step function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D left( lfloor left | x_i right | rfloor right) Equation: begin{equation} f(mathbf{x}) = sum_{i=1}^D left( lfloor left | x_i right | rfloor right) end{equation} Domain: -100 leq x_i leq 100 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Step benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Step'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of Step benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Step2(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Step2 function implementation. Date: 2018 Author: Lucija Brezočnik Licence: MIT Function: Step2 function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D left( lfloor x_i + 0.5 rfloor right)^2 Equation: begin{equation}f(mathbf{x}) = sum_{i=1}^D left( lfloor x_i + 0.5 rfloor right)^2 end{equation} Domain: -100 leq x_i leq 100 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Step2 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Step2'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of Step2 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Step3(Lower=- 100.0, Upper=100.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Step3 function implementation. Date: 2018 Author: Lucija Brezočnik Licence: MIT Function: Step3 function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D left( lfloor x_i^2 rfloor right) Equation: begin{equation}f(mathbf{x}) = sum_{i=1}^D left( lfloor x_i^2 rfloor right)end{equation} Domain: -100 leq x_i leq 100 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Step3 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Step3'] __init__(Lower=- 100.0, Upper=100.0)[source] Initialize of Step3 benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Stepint(Lower=- 5.12, Upper=5.12)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Stepint functions. Date: 2018 Author: Lucija Brezočnik License: MIT Function: Stepint function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D x_i^2 Equation: begin{equation}f(mathbf{x}) = sum_{i=1}^D x_i^2 end{equation} Domain: 0 leq x_i leq 10 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Stepint benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Stepint'] __init__(Lower=- 5.12, Upper=5.12)[source] Initialize of Stepint benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.StyblinskiTang(Lower=- 5.0, Upper=5.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Styblinski-Tang functions. Date: 2018 Authors: Lucija Brezočnik License: MIT Function: Styblinski-Tang function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = frac{1}{2} sum_{i=1}^D left( x_i^4 - 16x_i^2 + 5x_i right)  Equation: begin{equation}f(mathbf{x}) = frac{1}{2} sum_{i=1}^D left( x_i^4 - 16x_i^2 + 5x_i right) end{equation} Domain: -5 leq x_i leq 5 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Styblinski Tang benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['StyblinskiTang'] __init__(Lower=- 5.0, Upper=5.0)[source] Initialize of Styblinski Tang benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.SumSquares(Lower=- 10.0, Upper=10.0)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementation of Sum Squares functions. Date: 2018 Authors: Lucija Brezočnik License: MIT Function: Sum Squares function $$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)$$ LaTeX formats: Inline: f(mathbf{x}) = sum_{i=1}^D i x_i^2 Equation: begin{equation}f(mathbf{x}) = sum_{i=1}^D i x_i^2 end{equation} Domain: 0 leq x_i leq 10 Reference paper: Jamil, M., and Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. Initialize of Sum Squares benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['SumSquares'] __init__(Lower=- 10.0, Upper=10.0)[source] Initialize of Sum Squares benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Trid(D=2)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Trid functions. Date: 2018 Author: Klemen Berkovič License: MIT Function: Levy Function $$f(\textbf{x}) = \sum_{i = 1}^D \left( x_i - 1 \right)^2 - \sum_{i = 2}^D x_i x_{i - 1}$$ Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube $$x_i ∈ [-D^2, D^2]$$, for all $$i = 1, 2,..., D$$. Global minimum: $$f(\textbf{x}^*) = \frac{-D(D + 4)(D - 1)}{6}$$ at $$\textbf{x}^* = (1 (D + 1 - 1), \cdots , i (D + 1 - i) , \cdots , D (D + 1 - D))$$ LaTeX formats: Inline: f(textbf{x}) = sum_{i = 1}^D left( x_i - 1 right)^2 - sum_{i = 2}^D x_i x_{i - 1} Equation: begin{equation} f(textbf{x}) = sum_{i = 1}^D left( x_i - 1 right)^2 - sum_{i = 2}^D x_i x_{i - 1} end{equation} Domain: -D^2 leq x_i leq D^2 Reference: https://www.sfu.ca/~ssurjano/trid.html Initialize of Trid benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. Name = ['Trid'] __init__(D=2)[source] Initialize of Trid benchmark. Parameters • Lower (Optional[float]) – Lower bound of problem. • Upper (Optional[float]) – Upper bound of problem. function()[source] Return benchmark evaluation function. Returns Fitness function Return type Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float] static latex_code()[source] Return the latex code of the problem. Returns Latex code Return type str class NiaPy.benchmarks.Weierstrass(Lower=- 100.0, Upper=100.0, a=0.5, b=3, k_max=20)[source] Bases: NiaPy.benchmarks.benchmark.Benchmark Implementations of Weierstrass functions. Date: 2018 Author: Klemen Berkovič License: MIT Function: Weierstass Function $$f(\textbf{x}) = \sum_{i=1}^D \left( \sum_{k=0}^{k_{max}} a^k \cos\left( 2 \pi b^k ( x_i + 0.5) \right) \right) - D \sum_{k=0}^{k_{max}} a^k \cos \left( 2 \pi b^k \cdot 0.5 \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$$. Default value of a = 0.5, b = 3 and k_max = 20. Global minimum: $$f(x^*) = 0$$, at $$x^* = (420.968746,...,420.968746)$$ LaTeX formats: Inline:$$f(textbf{x}) = sum_{i=1}^D left( sum_{k=0}^{k_{max}} a^k cosleft( 2 pi b^k ( x_i + 0.5) right) right) - D sum_{k=0}^{k_{max}} a^k cos left( 2 pi b^k cdot 0.5 right)

Equation:

begin{equation} f(textbf{x}) = sum_{i=1}^D left( sum_{k=0}^{k_{max}} a^k cosleft( 2 pi b^k ( x_i + 0.5) right) right) - D sum_{k=0}^{k_{max}} a^k cos left( 2 pi b^k cdot 0.5 right) end{equation}

Domain:

$-100 leq x_i leq 100$

Reference:

Initialize of Bent Cigar benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

• a (Optional[float]) – TODO

• b (Optional[float]) – TODO

• k (Optional[float]) – TODO

Name = ['Weierstrass']
__init__(Lower=- 100.0, Upper=100.0, a=0.5, b=3, k_max=20)[source]

Initialize of Bent Cigar benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

• a (Optional[float]) – TODO

• b (Optional[float]) – TODO

• k (Optional[float]) – TODO

a = 0.5
b = 3
function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

k_max = 20
static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

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

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementation of Whitley function.

Date: 2018

Authors: Grega Vrbančič and Lucija Brezočnik

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.

Initialize of Whitley benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['Whitley']
__init__(Lower=- 10.24, Upper=10.24)[source]

Initialize of Whitley benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str

class NiaPy.benchmarks.Zakharov(Lower=- 5.0, Upper=10.0)[source]

Bases: NiaPy.benchmarks.benchmark.Benchmark

Implementations of Zakharov functions.

Date: 2018

Author: Klemen Berkovič

Function: Levy Function

$$f(\textbf{x}) = \sum_{i = 1}^D x_i^2 + \left( \sum_{i = 1}^D 0.5 i x_i \right)^2 + \left( \sum_{i = 1}^D 0.5 i x_i \right)^4$$

Input domain: The function can be defined on any input domain but it is usually evaluated on the hypercube $$x_i ∈ [-5, 10]$$, for all $$i = 1, 2,..., D$$.

Global minimum: $$f(\textbf{x}^*) = 0$$ at $$\textbf{x}^* = (0, \cdots, 0)$$

LaTeX formats:
Inline:

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

Equation:

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

Domain:

$-5 leq x_i leq 10$

Reference:

https://www.sfu.ca/~ssurjano/levy.html

Initialize of Zakharov benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

Name = ['Zakharov']
__init__(Lower=- 5.0, Upper=10.0)[source]

Initialize of Zakharov benchmark.

Parameters
• Lower (Optional[float]) – Lower bound of problem.

• Upper (Optional[float]) – Upper bound of problem.

function()[source]

Return benchmark evaluation function.

Returns

Fitness function

Return type

Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]

static latex_code()[source]

Return the latex code of the problem.

Returns

Latex code

Return type

str