N-D Test Functions K¶

class Katsuura(dimensions=2)¶

Katsuura objective function.

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

$f_{\text{Katsuura}}(x) = \prod_{i=0}^{n-1} \left [ 1 + (i+1) \sum_{k=1}^{d} \lfloor (2^k x_i) \rfloor 2^{-k} \right ]$

Where, in this exercise, $d = 32$ .

Here, $n$ represents the number of dimensions and $x_i \in [0, 100]$ for $i = 1, ..., n$ .

Two-dimensional Katsuura function

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

Adorio, E. MVF - “Multivariate Test Functions Library in C for Unconstrained Global Optimization”, 2005 Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

Adorio has wrong global minimum. Adorio uses round, Gavana docstring uses floor, but Gavana code uses round. We’ll use round...

class Keane(dimensions=2)¶

Keane objective function.

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

$f_{\text{Keane}}(x) = \frac{\sin^2(x_1 - x_2)\sin^2(x_1 + x_2)} {\sqrt{x_1^2 + x_2^2}}$

with $x_i \in [0, 10]$ for $i = 1, 2$ .

Two-dimensional Keane function

Global optimum: $f(x) = 0.0$ for $x = [7.85396153, 7.85396135]$ .

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

Jamil #69, there is no way that the function can have a negative value. Everything is squared. I think that they have the wrong solution.

class Kearfott(dimensions=2)¶: Kearfott Cosine objective function.

Two-dimensional Kearfott function

class Kowalik(dimensions=4)¶

Kowalik objective function.

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

$f_{\text{Kowalik}}(x) = \sum_{i=0}^{10} \left [ a_i - \frac{x_1 (b_i^2 + b_i x_2)} {b_i^2 + b_i x_3 + x_4} \right ]^2$

Where:

$\begin{matrix} a = [4, 2, 1, 1/2, 1/4 1/8, 1/10, 1/12, 1/14, 1/16] \\ b = [0.1957, 0.1947, 0.1735, 0.1600, 0.0844, 0.0627, 0.0456, 0.0342, 0.0323, 0.0235, 0.0246] \\ \end{matrix}$

Here, $n$ represents the number of dimensions and $x_i \in [-5, 5]$ for $i = 1, ..., 4$ .

Global optimum: $f(x) = 0.00030748610$ for $x = [0.192833, 0.190836, 0.123117, 0.135766]$ .

http://www.itl.nist.gov/div898/strd/nls/data/mgh09.shtml

N-D Test Functions K¶

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