test_functions 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.

Katsuura function

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.

Keane function

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.

Kearfott 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

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