N-D Test Functions G¶

class Gear(dimensions=4)¶

Gear objective function.

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

$f_{\text{Gear}}({x}) = \left \{ \frac{1.0}{6.931} - \frac{\lfloor x_1\rfloor \lfloor x_2 \rfloor } {\lfloor x_3 \rfloor \lfloor x_4 \rfloor } \right\}^2$

with $x_i \in [12, 60]$ for $i = 1, ..., 4$ .

Global optimum: $f(x) = 2.7 \cdot 10^{-12}$ for $x = [16, 19, 43, 49]$ , where the various $x_i$ may be permuted.

Gavana, A. Global Optimization Benchmarks and AMPGO

class Giunta(dimensions=2)¶

Giunta objective function.

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

$f_{\text{Giunta}}({x}) = 0.6 + \sum_{i=1}^{n} \left[\sin^{2}\left(1 - \frac{16}{15} x_i\right) - \frac{1}{50} \sin\left(4 - \frac{64}{15} x_i\right) - \sin\left(1 - \frac{16}{15} x_i\right)\right]$

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

Two-dimensional Giunta function

Global optimum: $f(x) = 0.06447042053690566$ for $x = [0.4673200277395354, 0.4673200169591304]$

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 has the wrong fglob. I think there is a lower value.

class GoldsteinPrice(dimensions=2)¶

Goldstein-Price objective function.

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

$f_{\text{GoldsteinPrice}}(x) = \left[ 1 + (x_1 + x_2 + 1)^2 (19 - 14 x_1 + 3 x_1^2 - 14 x_2 + 6 x_1 x_2 + 3 x_2^2) \right] \left[ 30 + ( 2x_1 - 3 x_2)^2 (18 - 32 x_1 + 12 x_1^2 + 48 x_2 - 36 x_1 x_2 + 27 x_2^2) \right]$

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

Two-dimensional GoldsteinPrice function

Global optimum: $f(x) = 3$ for $x = [0, -1]$

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 GramacyLee02(dimensions=2)¶: Gramacy Lee objective function No 2.

Two-dimensional GramacyLee02 function

class GramacyLee03(dimensions=2)¶: Gramacy Lee objective function No 3.

Two-dimensional GramacyLee03 function

class Griewank(dimensions=2)¶

Griewank objective function.

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

$f_{\text{Griewank}}(x) = \frac{1}{4000}\sum_{i=1}^n x_i^2 - \prod_{i=1}^n\cos\left(\frac{x_i}{\sqrt{i}}\right) + 1$

Here, $n$ represents the number of dimensions and $x_i \in [-600, 600]$ for $i = 1, ..., n$ .

Two-dimensional Griewank 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 Gulf(dimensions=3)¶

Gulf objective function.

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

$f_{\text{Gulf}}(x) = \sum_{i=1}^99 \left( e^{-\frac{\lvert y_i - x_2 \rvert^{x_3}}{x_1}} - t_i \right)$

Where, in this exercise:

$t_i = i/100 \ y_i = 25 + [-50 \log(t_i)]^{2/3}$

with $x_i \in [0, 60]$ for $i = 1, 2, 3$ .

Global optimum: $f(x) = 0$ for $x = [50, 25, 1.5]$

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

Gavana has absolute of (u - x[1]) term. Jamil doesn’t... Leaving it in.

N-D Test Functions G¶

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