Benchmarks

Regroup typical EC benchmarks functions to import easily and benchmark examples.

Single Objective Continuous	Multi Objective Continuous	Binary	Symbolic Regression
cigar()	fonseca()	chuang_f1()	kotanchek()
plane()	kursawe()	chuang_f2()	salustowicz_1d()
sphere()	schaffer_mo()	chuang_f3()	salustowicz_2d()
rand()	dtlz1()	royal_road1()	unwrapped_ball()
ackley()	dtlz2()	royal_road2()	rational_polynomial()
bohachevsky()	dtlz3()		rational_polynomial2()
griewank()	dtlz4()		sin_cos()
h1()	zdt1()		ripple()
himmelblau()	zdt2()
rastrigin()	zdt3()
rastrigin_scaled()	zdt4()
rastrigin_skew()	zdt6()
rosenbrock()
schaffer()
schwefel()
shekel()

Continuous Optimization

deap.benchmarks.cigar(individual)

Cigar test objective function.

Type	minimization
Range	none
Global optima	\(x_i = 0, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\(f(\mathbf{x}) = x_0^2 + 10^6\sum_{i=1}^N\,x_i^2\)

deap.benchmarks.plane(individual)

Plane test objective function.

Type	minimization
Range	none
Global optima	\(x_i = 0, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\(f(\mathbf{x}) = x_0\)

deap.benchmarks.sphere(individual)

Sphere test objective function.

Type	minimization
Range	none
Global optima	\(x_i = 0, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\(f(\mathbf{x}) = \sum_{i=1}^Nx_i^2\)

deap.benchmarks.rand(individual)

Random test objective function.

Type	minimization or maximization
Range	none
Global optima	none
Function	\(f(\mathbf{x}) = \text{\texttt{random}}(0,1)\)

deap.benchmarks.ackley(individual)

Ackley test objective function.

Type	minimization
Range	\(x_i \in [-15, 30]\)
Global optima	\(x_i = 0, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\[f(\mathbf{x}) = 20 - 20\exp\left(-0.2\sqrt{\frac{1}{N} \sum_{i=1}^N x_i^2} \right) + e - \exp\left(\frac{1}{N}\sum_{i=1}^N \cos(2\pi x_i) \right)\]

deap.benchmarks.bohachevsky(individual)

Bohachevsky test objective function.

Type	minimization
Range	\(x_i \in [-100, 100]\)
Global optima	\(x_i = 0, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\[f(\mathbf{x}) = \sum_{i=1}^{N-1}(x_i^2 + 2x_{i+1}^2 - 0.3\cos(3\pi x_i) - 0.4\cos(4\pi x_{i+1}) + 0.7)\]

deap.benchmarks.griewank(individual)

Griewank test objective function.

Type	minimization
Range	\(x_i \in [-600, 600]\)
Global optima	\(x_i = 0, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\[f(\mathbf{x}) = \frac{1}{4000}\sum_{i=1}^N\,x_i^2 - \prod_{i=1}^N\cos\left(\frac{x_i}{\sqrt{i}}\right) + 1\]

deap.benchmarks.h1(individual)

Simple two-dimensional function containing several local maxima. From: The Merits of a Parallel Genetic Algorithm in Solving Hard Optimization Problems, A. J. Knoek van Soest and L. J. R. Richard Casius, J. Biomech. Eng. 125, 141 (2003)

Type	maximization
Range	\(x_i \in [-100, 100]\)
Global optima	\(\mathbf{x} = (8.6998, 6.7665)\), \(f(\mathbf{x}) = 2\)n
Function	\[f(\mathbf{x}) = \frac{\sin(x_1 - \frac{x_2}{8})^2 + \ \sin(x_2 + \frac{x_1}{8})^2}{\sqrt{(x_1 - 8.6998)^2 + \ (x_2 - 6.7665)^2} + 1}\]

deap.benchmarks.himmelblau(individual)

The Himmelblau’s function is multimodal with 4 defined minimums in \([-6, 6]^2\).

Type	minimization
Range	\(x_i \in [-6, 6]\)
Global optima	\(\mathbf{x}_1 = (3.0, 2.0)\), \(f(\mathbf{x}_1) = 0\)n \(\mathbf{x}_2 = (-2.805118, 3.131312)\), \(f(\mathbf{x}_2) = 0\)n \(\mathbf{x}_3 = (-3.779310, -3.283186)\), \(f(\mathbf{x}_3) = 0\)n \(\mathbf{x}_4 = (3.584428, -1.848126)\), \(f(\mathbf{x}_4) = 0\)n
Function	\(f(x_1, x_2) = (x_1^2 + x_2 - 11)^2 + (x_1 + x_2^2 -7)^2\)

deap.benchmarks.rastrigin(individual)

Rastrigin test objective function.

Type	minimization
Range	\(x_i \in [-5.12, 5.12]\)
Global optima	\(x_i = 0, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\(f(\mathbf{x}) = 10N + \sum_{i=1}^N x_i^2 - 10 \cos(2\pi x_i)\)

deap.benchmarks.rastrigin_scaled(individual)

Scaled Rastrigin test objective function.

\[f_{\text{RastScaled}}(\mathbf{x}) = 10N + \sum_{i=1}^N \left(10^{\left(\frac{i-1}{N-1}\right)} x_i \right)^2 - 10\cos\left(2\pi 10^{\left(\frac{i-1}{N-1}\right)} x_i \right)`\]

deap.benchmarks.rastrigin_skew(individual)

Skewed Rastrigin test objective function.

\[f_{\text{RastSkew}}(\mathbf{x}) = 10N + \sum_{i=1}^N \left(y_i^2 - 10 \cos(2\pi x_i)\right)\]

\[\begin{split}\text{with } y_i = \begin{cases} 10\cdot x_i & \text{ if } x_i > 0,\\ \\ x_i & \text{ otherwise } \end{cases}\end{split}\]

deap.benchmarks.rosenbrock(individual)

Rosenbrock test objective function.

Type	minimization
Range	none
Global optima	\(x_i = 1, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\(f(\mathbf{x}) = \sum_{i=1}^{N-1} (1-x_i)^2 + 100 (x_{i+1} - x_i^2 )^2\)

deap.benchmarks.schaffer(individual)

Schaffer test objective function.

Type	minimization
Range	\(x_i \in [-100, 100]\)
Global optima	\(x_i = 0, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\[f(\mathbf{x}) = \sum_{i=1}^{N-1} (x_i^2+x_{i+1}^2)^{0.25} \cdot \left[ \sin^2(50\cdot(x_i^2+x_{i+1}^2)^{0.10}) + 1.0 \right]\]

deap.benchmarks.schwefel(individual)

Schwefel test objective function.

Type	minimization
Range	\(x_i \in [-500, 500]\)
Global optima	\(x_i = 420.96874636, \forall i \in \lbrace 1 \ldots N\rbrace\), \(f(\mathbf{x}) = 0\)
Function	\[f(\mathbf{x}) = 418.9828872724339\cdot N - \sum_{i=1}^N\,x_i\sin\left(\sqrt{|x_i|}\right)\]

deap.benchmarks.shekel(individual, a, c)

The Shekel multimodal function can have any number of maxima. The number of maxima is given by the length of any of the arguments a or c, a is a matrix of size \(M\times N\), where M is the number of maxima and N the number of dimensions and c is a \(M\times 1\) vector.

\(f_\text{Shekel}(\mathbf{x}) = \sum_{i = 1}^{M} \frac{1}{c_{i} + \sum_{j = 1}^{N} (x_{j} - a_{ij})^2 }\)

The following figure uses

\(\mathcal{A} = \begin{bmatrix} 0.5 & 0.5 \\ 0.25 & 0.25 \\ 0.25 & 0.75 \\ 0.75 & 0.25 \\ 0.75 & 0.75 \end{bmatrix}\) and \(\mathbf{c} = \begin{bmatrix} 0.002 \\ 0.005 \\ 0.005 \\ 0.005 \\ 0.005 \end{bmatrix}\), thus defining 5 maximums in \(\mathbb{R}^2\).

Multi-objective

deap.benchmarks.fonseca(individual)

Fonseca and Fleming’s multiobjective function. From: C. M. Fonseca and P. J. Fleming, “Multiobjective optimization and multiple constraint handling with evolutionary algorithms – Part II: Application example”, IEEE Transactions on Systems, Man and Cybernetics, 1998.

\(f_{\text{Fonseca}1}(\mathbf{x}) = 1 - e^{-\sum_{i=1}^{3}(x_i - \frac{1}{\sqrt{3}})^2}\)

\(f_{\text{Fonseca}2}(\mathbf{x}) = 1 - e^{-\sum_{i=1}^{3}(x_i + \frac{1}{\sqrt{3}})^2}\)

deap.benchmarks.kursawe(individual)

Kursawe multiobjective function.

\(f_{\text{Kursawe}1}(\mathbf{x}) = \sum_{i=1}^{N-1} -10 e^{-0.2 \sqrt{x_i^2 + x_{i+1}^2} }\)

\(f_{\text{Kursawe}2}(\mathbf{x}) = \sum_{i=1}^{N} |x_i|^{0.8} + 5 \sin(x_i^3)\)

deap.benchmarks.schaffer_mo(individual)

Schaffer’s multiobjective function on a one attribute individual. From: J. D. Schaffer, “Multiple objective optimization with vector evaluated genetic algorithms”, in Proceedings of the First International Conference on Genetic Algorithms, 1987.

\(f_{\text{Schaffer}1}(\mathbf{x}) = x_1^2\)

\(f_{\text{Schaffer}2}(\mathbf{x}) = (x_1-2)^2\)

deap.benchmarks.dtlz1(individual, obj)

DTLZ1 multiobjective function. It returns a tuple of obj values. The individual must have at least obj elements. From: K. Deb, L. Thiele, M. Laumanns and E. Zitzler. Scalable Multi-Objective Optimization Test Problems. CEC 2002, p. 825 - 830, IEEE Press, 2002.

\(g(\mathbf{x}_m) = 100\left(|\mathbf{x}_m| + \sum_{x_i \in \mathbf{x}_m}\left((x_i - 0.5)^2 - \cos(20\pi(x_i - 0.5))\right)\right)\)

\(f_{\text{DTLZ1}1}(\mathbf{x}) = \frac{1}{2} (1 + g(\mathbf{x}_m)) \prod_{i=1}^{m-1}x_i\)

\(f_{\text{DTLZ1}2}(\mathbf{x}) = \frac{1}{2} (1 + g(\mathbf{x}_m)) (1-x_{m-1}) \prod_{i=1}^{m-2}x_i\)

\(\ldots\)

\(f_{\text{DTLZ1}m-1}(\mathbf{x}) = \frac{1}{2} (1 + g(\mathbf{x}_m)) (1 - x_2) x_1\)

\(f_{\text{DTLZ1}m}(\mathbf{x}) = \frac{1}{2} (1 - x_1)(1 + g(\mathbf{x}_m))\)

Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.

deap.benchmarks.dtlz2(individual, obj)

DTLZ2 multiobjective function. It returns a tuple of obj values. The individual must have at least obj elements. From: K. Deb, L. Thiele, M. Laumanns and E. Zitzler. Scalable Multi-Objective Optimization Test Problems. CEC 2002, p. 825 - 830, IEEE Press, 2002.

\(g(\mathbf{x}_m) = \sum_{x_i \in \mathbf{x}_m} (x_i - 0.5)^2\)

\(f_{\text{DTLZ2}1}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \prod_{i=1}^{m-1} \cos(0.5x_i\pi)\)

\(f_{\text{DTLZ2}2}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{m-1}\pi ) \prod_{i=1}^{m-2} \cos(0.5x_i\pi)\)

\(\ldots\)

\(f_{\text{DTLZ2}m}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{1}\pi )\)

Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.

deap.benchmarks.dtlz3(individual, obj)

DTLZ3 multiobjective function. It returns a tuple of obj values. The individual must have at least obj elements. From: K. Deb, L. Thiele, M. Laumanns and E. Zitzler. Scalable Multi-Objective Optimization Test Problems. CEC 2002, p. 825 - 830, IEEE Press, 2002.

\(g(\mathbf{x}_m) = 100\left(|\mathbf{x}_m| + \sum_{x_i \in \mathbf{x}_m}\left((x_i - 0.5)^2 - \cos(20\pi(x_i - 0.5))\right)\right)\)

\(f_{\text{DTLZ3}1}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \prod_{i=1}^{m-1} \cos(0.5x_i\pi)\)

\(f_{\text{DTLZ3}2}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{m-1}\pi ) \prod_{i=1}^{m-2} \cos(0.5x_i\pi)\)

\(\ldots\)

\(f_{\text{DTLZ3}m}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{1}\pi )\)

Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.

deap.benchmarks.dtlz4(individual, obj, alpha)

DTLZ4 multiobjective function. It returns a tuple of obj values. The individual must have at least obj elements. The alpha parameter allows for a meta-variable mapping in dtlz2() \(x_i \rightarrow x_i^\alpha\), the authors suggest \(\alpha = 100\). From: K. Deb, L. Thiele, M. Laumanns and E. Zitzler. Scalable Multi-Objective Optimization Test Problems. CEC 2002, p. 825 - 830, IEEE Press, 2002.

\(g(\mathbf{x}_m) = \sum_{x_i \in \mathbf{x}_m} (x_i - 0.5)^2\)

\(f_{\text{DTLZ4}1}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \prod_{i=1}^{m-1} \cos(0.5x_i^\alpha\pi)\)

\(f_{\text{DTLZ4}2}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{m-1}^\alpha\pi ) \prod_{i=1}^{m-2} \cos(0.5x_i^\alpha\pi)\)

\(\ldots\)

\(f_{\text{DTLZ4}m}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{1}^\alpha\pi )\)

Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.

deap.benchmarks.zdt1(individual)

ZDT1 multiobjective function.

\(g(\mathbf{x}) = 1 + \frac{9}{n-1}\sum_{i=2}^n x_i\)

\(f_{\text{ZDT1}1}(\mathbf{x}) = x_1\)

\(f_{\text{ZDT1}2}(\mathbf{x}) = g(\mathbf{x})\left[1 - \sqrt{\frac{x_1}{g(\mathbf{x})}}\right]\)

deap.benchmarks.zdt2(individual)

ZDT2 multiobjective function.

\(g(\mathbf{x}) = 1 + \frac{9}{n-1}\sum_{i=2}^n x_i\)

\(f_{\text{ZDT2}1}(\mathbf{x}) = x_1\)

\(f_{\text{ZDT2}2}(\mathbf{x}) = g(\mathbf{x})\left[1 - \left(\frac{x_1}{g(\mathbf{x})}\right)^2\right]\)

deap.benchmarks.zdt3(individual)

ZDT3 multiobjective function.

\(g(\mathbf{x}) = 1 + \frac{9}{n-1}\sum_{i=2}^n x_i\)

\(f_{\text{ZDT3}1}(\mathbf{x}) = x_1\)

\(f_{\text{ZDT3}2}(\mathbf{x}) = g(\mathbf{x})\left[1 - \sqrt{\frac{x_1}{g(\mathbf{x})}} - \frac{x_1}{g(\mathbf{x})}\sin(10\pi x_1)\right]\)

deap.benchmarks.zdt4(individual)

ZDT4 multiobjective function.

\(g(\mathbf{x}) = 1 + 10(n-1) + \sum_{i=2}^n \left[ x_i^2 - 10\cos(4\pi x_i) \right]\)

\(f_{\text{ZDT4}1}(\mathbf{x}) = x_1\)

\(f_{\text{ZDT4}2}(\mathbf{x}) = g(\mathbf{x})\left[ 1 - \sqrt{x_1/g(\mathbf{x})} \right]\)

deap.benchmarks.zdt6(individual)

ZDT6 multiobjective function.

\(g(\mathbf{x}) = 1 + 9 \left[ \left(\sum_{i=2}^n x_i\right)/(n-1) \right]^{0.25}\)

\(f_{\text{ZDT6}1}(\mathbf{x}) = 1 - \exp(-4x_1)\sin^6(6\pi x_1)\)

\(f_{\text{ZDT6}2}(\mathbf{x}) = g(\mathbf{x}) \left[ 1 - (f_{\text{ZDT6}1}(\mathbf{x})/g(\mathbf{x}))^2 \right]\)

Binary Optimization

deap.benchmarks.binary.chuang_f1(individual)

Binary deceptive function from : Multivariate Multi-Model Approach for Globally Multimodal Problems by Chung-Yao Chuang and Wen-Lian Hsu.

The function takes individual of 40+1 dimensions and has two global optima in [1,1,…,1] and [0,0,…,0].

deap.benchmarks.binary.chuang_f2(individual)

Binary deceptive function from : Multivariate Multi-Model Approach for Globally Multimodal Problems by Chung-Yao Chuang and Wen-Lian Hsu.

The function takes individual of 40+1 dimensions and has four global optima in [1,1,…,0,0], [0,0,…,1,1], [1,1,…,1] and [0,0,…,0].

deap.benchmarks.binary.chuang_f3(individual)

Binary deceptive function from : Multivariate Multi-Model Approach for Globally Multimodal Problems by Chung-Yao Chuang and Wen-Lian Hsu.

The function takes individual of 40+1 dimensions and has two global optima in [1,1,…,1] and [0,0,…,0].

deap.benchmarks.binary.royal_road1(individual, order)

Royal Road Function R1 as presented by Melanie Mitchell in : “An introduction to Genetic Algorithms”.

deap.benchmarks.binary.royal_road2(individual, order)

Royal Road Function R2 as presented by Melanie Mitchell in : “An introduction to Genetic Algorithms”.

deap.benchmarks.binary.bin2float(min_, max_, nbits)

Convert a binary array into an array of float where each float is composed of nbits and is between min_ and max_ and return the result of the decorated function.

Symbolic Regression

deap.benchmarks.gp.kotanchek(data)

Kotanchek benchmark function.

Range	\(\mathbf{x} \in [-1, 7]^2\)
Function	\(f(\mathbf{x}) = \\frac{e^{-(x_1 - 1)^2}}{3.2 + (x_2 - 2.5)^2}\)

deap.benchmarks.gp.salustowicz_1d(data)

Salustowicz benchmark function.

Range	\(x \in [0, 10]\)
Function	\(f(x) = e^{-x} x^3 \cos(x) \sin(x) (\cos(x) \sin^2(x) - 1)\)

deap.benchmarks.gp.salustowicz_2d(data)

Salustowicz benchmark function.

Range	\(\mathbf{x} \in [0, 7]^2\)
Function	\(f(\mathbf{x}) = e^{-x_1} x_1^3 \cos(x_1) \sin(x_1) (\cos(x_1) \sin^2(x_1) - 1) (x_2 -5)\)

deap.benchmarks.gp.unwrapped_ball(data)

Unwrapped ball benchmark function.

Range	\(\mathbf{x} \in [-2, 8]^n\)
Function	\(f(\mathbf{x}) = \\frac{10}{5 + \sum_{i=1}^n (x_i - 3)^2}\)

deap.benchmarks.gp.rational_polynomial(data)

Rational polynomial ball benchmark function.

Range	\(\mathbf{x} \in [0, 2]^3\)
Function	\(f(\mathbf{x}) = \\frac{30 * (x_1 - 1) (x_3 - 1)}{x_2^2 (x_1 - 10)}\)

deap.benchmarks.gp.rational_polynomial2(data)

Rational polynomial benchmark function.

Range	\(\mathbf{x} \in [0, 6]^2\)
Function	\(f(\mathbf{x}) = \\frac{(x_1 - 3)^4 + (x_2 - 3)^3 - (x_2 - 3)}{(x_2 - 2)^4 + 10}\)

deap.benchmarks.gp.sin_cos(data)

Sine cosine benchmark function.

Range	\(\mathbf{x} \in [0, 6]^2\)
Function	\(f(\mathbf{x}) = 6\sin(x_1)\cos(x_2)\)

deap.benchmarks.gp.ripple(data)

Ripple benchmark function.

Range	\(\mathbf{x} \in [-5, 5]^2\)
Function	\(f(\mathbf{x}) = (x_1 - 3) (x_2 - 3) + 2 \sin((x_1 - 4) (x_2 -4))\)

Moving Peaks Benchmark

Re-implementation of the Moving Peaks Benchmark by Jurgen Branke. With the addition of the fluctuating number of peaks presented in du Plessis and Engelbrecht, 2013, Self-Adaptive Environment with Fluctuating Number of Optima.

class deap.benchmarks.movingpeaks.MovingPeaks(self, dim[, pfunc][, npeaks][, bfunc][, random][, ...])

The Moving Peaks Benchmark is a fitness function changing over time. It consists of a number of peaks, changing in height, width and location. The peaks function is given by pfunc, which is either a function object or a list of function objects (the default is function1()). The number of peaks is determined by npeaks (which defaults to 5). This parameter can be either a integer or a sequence. If it is set to an integer the number of peaks won’t change, while if set to a sequence of 3 elements, the number of peaks will fluctuate between the first and third element of that sequence, the second element is the initial number of peaks. When fluctuating the number of peaks, the parameter number_severity must be included, it represents the number of peak fraction that is allowed to change. The dimensionality of the search domain is dim. A basis function bfunc can also be given to act as static landscape (the default is no basis function). The argument random serves to grant an independent random number generator to the moving peaks so that the evolution is not influenced by number drawn by this object (the default uses random functions from the Python module random). Various other keyword parameters listed in the table below are required to setup the benchmark, default parameters are based on scenario 1 of this benchmark.

Parameter	SCENARIO_1 (Default)	SCENARIO_2	SCENARIO_3	Details
pfunc	function1()	cone()	cone()	The peak function or a list of peak function.
npeaks	5	10	50	Number of peaks. If an integer, the number of peaks won’t change, if a sequence it will fluctuate [min, current, max].
bfunc	None	None	lambda x: 10	Basis static function.
min_coord	0.0	0.0	0.0	Minimum coordinate for the centre of the peaks.
max_coord	100.0	100.0	100.0	Maximum coordinate for the centre of the peaks.
min_height	30.0	30.0	30.0	Minimum height of the peaks.
max_height	70.0	70.0	70.0	Maximum height of the peaks.
uniform_height	50.0	50.0	0	Starting height for all peaks, if uniform_height <= 0 the initial height is set randomly for each peak.
min_width	0.0001	1.0	1.0	Minimum width of the peaks.
max_width	0.2	12.0	12.0	Maximum width of the peaks
uniform_width	0.1	0	0	Starting width for all peaks, if uniform_width <= 0 the initial width is set randomly for each peak.
lambda_	0.0	0.5	0.5	Correlation between changes.
move_severity	1.0	1.5	1.0	The distance a single peak moves when peaks change.
height_severity	7.0	7.0	1.0	The standard deviation of the change made to the height of a peak when peaks change.
width_severity	0.01	1.0	0.5	The standard deviation of the change made to the width of a peak when peaks change.
period	5000	5000	1000	Period between two changes.

Dictionaries SCENARIO_1, SCENARIO_2 and SCENARIO_3 of this module define the defaults for these parameters. The scenario 3 requires a constant basis function which can be given as a lambda function lambda x: constant.

The following shows an example of scenario 1 with non uniform heights and widths.

__call__(self, individual[, count])

Evaluate a given individual with the current benchmark configuration.

Parameters:

• indidivudal – The individual to evaluate.

• count – Whether or not to count this evaluation in the total evaluation count. (Defaults to True)

changePeaks()

Order the peaks to change position, height, width and number.

globalMaximum()

Returns the global maximum value and position.

maximums()

Returns all visible maximums value and position sorted with the global maximum first.

deap.benchmarks.movingpeaks.cone(individual, position, height, width)

The cone peak function to be used with scenario 2 and 3.

\(f(\mathbf{x}) = h - w \sqrt{\sum_{i=1}^N (x_i - p_i)^2}\)

deap.benchmarks.movingpeaks.function1(individual, position, height, width)

The function1 peak function to be used with scenario 1.

\(f(\mathbf{x}) = \\frac{h}{1 + w \sqrt{\sum_{i=1}^N (x_i - p_i)^2}}\)

Benchmarks tools