N-D Test Functions W¶

class Watson(dimensions=6)¶

Watson objective function.

This class defines the Watson global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{Watson}}(x) = \sum_{i=0}^{29} \left\{ \sum_{j=0}^4 ((j + 1)a_i^j x_{j+1}) - \left[ \sum_{j=0}^5 a_i^j x_{j+1} \right ]^2 - 1 \right\}^2 + x_1^2$

Where, in this exercise, $a_i = i/29$ .

with $x_i \in [-5, 5]$ for $i = 1, ..., 6$ .

Global optimum: $f(x) = 0.002288$ for $x = [-0.0158, 1.012, -0.2329, 1.260, -1.513, 0.9928]$

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 #161 writes equation using (j - 1). According to code in Adorio and Gavana it should be (j+1). However the equations in those papers contain (j - 1) as well. However, I’ve got the right global minimum!!!

class Wavy(dimensions=2)¶

Wavy objective function.

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

$f_{\text{Wavy}}(x) = 1 - \frac{1}{n} \sum_{i=1}^{n} \cos(kx_i)e^{-\frac{x_i^2}{2}}$

Where, in this exercise, $k = 10$ . The number of local minima is $kn$ and $(k + 1)n$ for odd and even $k$ respectively.

Here, $x_i \in [-\pi, \pi]$ for $i = 1, 2$ .

Two-dimensional Wavy function

Global optimum: $f(x) = 0$ 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 WayburnSeader01(dimensions=2)¶

Wayburn and Seader 1 objective function.

This class defines the Wayburn and Seader 1 global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{WayburnSeader01}}(x) = (x_1^6 + x_2^4 - 17)^2 + (2x_1 + x_2 - 4)^2$

with $x_i \in [-5, 5]$ for $i = 1, 2$ .

Two-dimensional WayburnSeader01 function

Global optimum: $f(x) = 0$ for $x = [1, 2]$

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.

class WayburnSeader02(dimensions=2)¶

Wayburn and Seader 2 objective function.

This class defines the Wayburn and Seader 2 global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{WayburnSeader02}}(x) = \left[ 1.613 - 4(x_1 - 0.3125)^2 - 4(x_2 - 1.625)^2 \right]^2 + (x_2 - 1)^2$

with $x_i \in [-500, 500]$ for $i = 1, 2$ .

Two-dimensional WayburnSeader02 function

Global optimum: $f(x) = 0$ for $x = [0.2, 1]$

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 WayburnSeader03(dimensions=2)¶: Wayburn and Seader 3 objective function.

Two-dimensional WayburnSeader03 function

class Weierstrass(dimensions=2)¶

Weierstrass objective function.

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

$f_{\text{Weierstrass}}(x) = \sum_{i=1}^{n} \left [ \sum_{k=0}^{kmax} a^k \cos \left( 2 \pi b^k (x_i + 0.5) \right) - n \sum_{k=0}^{kmax} a^k \cos(\pi b^k) \right ]$

Where, in this exercise, $kmax = 20$ , $a = 0.5$ and $b = 3$ .

Here, $n$ represents the number of dimensions and $x_i \in [-0.5, 0.5]$ for $i = 1, ..., n$ .

Two-dimensional Weierstrass function

Global optimum: $f(x) = 4$ for $x_i = 0$ for $i = 1, ..., n$

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

Todo

Jamil, Gavana have got it wrong. The second term is not supposed to be included in the outer sum. Mishra code has it right as does the reference referred to in Jamil#166.

class Whitley(dimensions=2)¶

Whitley objective function.

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

$f_{\text{Whitley}}(x) = \sum_{i=1}^n \sum_{j=1}^n \left[\frac{(100(x_i^2-x_j)^2 + (1-x_j)^2)^2}{4000} - \cos(100(x_i^2-x_j)^2 + (1-x_j)^2)+1 \right]$

Here, $n$ represents the number of dimensions and $x_i \in [-10.24, 10.24]$ for $i = 1, ..., n$ .

Two-dimensional Whitley function

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

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

Jamil#167 has ‘+ 1’ inside the cos term, when it should be outside it.

class Wolfe(dimensions=3)¶

Wolfe objective function.

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

$f_{\text{Wolfe}}(x) = \frac{4}{3}(x_1^2 + x_2^2 - x_1x_2)^{0.75} + x_3$

with $x_i \in [0, 2]$ for $i = 1, 2, 3$ .

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

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.

N-D Test Functions W¶

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