N-D Test Functions A¶

class Ackley01(dimensions=2)¶

Ackley01 objective function.

The Ackley01 global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Ackley01}}(x) = -20 e^{-0.2 \sqrt{\frac{1}{n} \sum_{i=1}^n x_i^2}} - e^{\frac{1}{n} \sum_{i=1}^n \cos(2 \pi x_i)} + 20 + e$

Here, $n$ represents the number of dimensions and $x_i \in [-35, 35]$ for $i = 1, ..., n$ .

Two-dimensional Ackley01 function

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

Adorio, E. MVF - “Multivariate Test Functions Library in C for Unconstrained Global Optimization”, 2005

Todo

the -0.2 factor in the exponent of the first term is given as -0.02 in Jamil et al.

class Ackley02(dimensions=2)¶

Ackley02 objective function.

The Ackley02 global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Ackley02}}(x) = -200 e^{-0.02 \sqrt{x_1^2 + x_2^2}}$

with $x_i \in [-32, 32]$ for $i=1, 2$ .

Two-dimensional Ackley02 function

Global optimum: $f(x) = -200$ for $x = [0, 0]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Ackley03(dimensions=2)¶

Ackley03 objective function.

The Ackley03 global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Ackley03}}(x) = -200 e^{-0.02 \sqrt{x_1^2 + x_2^2}} + 5e^{\cos(3x_1) + \sin(3x_2)}$

with $x_i \in [-32, 32]$ for $i=1, 2$ .

Two-dimensional Ackley03 function

Global optimum: $f(x) = -195.62902825923879$ for $x = [-0.68255758, -0.36070859]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

I think the minus sign is missing in front of the first term in eqn3 in Jamil. This changes the global minimum

class Ackley04(dimensions=2)¶: Ackley04 objective function.

Two-dimensional Ackley04 function

class Adjiman(dimensions=2)¶

Adjiman objective function.

The Adjiman global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Adjiman}}(x) = \cos(x_1)\sin(x_2) - \frac{x_1}{(x_2^2 + 1)}$

with, $x_1\in [-1, 2]$ and $x_2\in [-1, 1]$ .

Two-dimensional Adjiman function

Global optimum: $f(x) = -2.02181$ for $x = [2.0, 0.10578]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Alpine01(dimensions=2)¶

Alpine01 objective function.

The Alpine01 global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Alpine01}}(x) = \sum_{i=1}^{n} \lvert {x_i \sin \left( x_i \right) + 0.1 x_i} \rvert$

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

Two-dimensional Alpine01 function

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

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Alpine02(dimensions=2)¶

Alpine02 objective function.

The Alpine02 global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Alpine02}}(x) = \prod_{i=1}^{n} \sqrt{x_i} \sin(x_i)$

Here, $n$ represents the number of dimensions and $x_i \in [0, 10]$ for $i = 1, ..., n$ .

Two-dimensional Alpine02 function

Global optimum: $f(x) = -6.12950389$ for $x = [7.91705268, 4.81584232]$ for $i = 1, 2$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

eqn 7 in has the wrong global minimum value.

class AMGM(dimensions=2)¶

AMGM objective function.

The AMGM (Arithmetic Mean - Geometric Mean Equality) global optimization problem is a multimodal minimization problem defined as follows

$f_{\text{AMGM}}(x) = \left ( \frac{1}{n} \sum_{i=1}^{n} x_i - \sqrt[n]{ \prod_{i=1}^{n} x_i} \right )^2$

Here, $n$ represents the number of dimensions and $x_i \in [0, 10]$ for $i = 1, ..., n$ .

Two-dimensional AMGM function

Global optimum: $f(x) = 0$ for $x_1 = x_2 = ... = x_n$ for $i = 1, ..., n$

Gavana, A. Global Optimization Benchmarks and AMPGO

N-D Test Functions A¶

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