N-D Test Functions N¶

class NeedleEye(dimensions=2)¶

NeedleEye objective function.

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

$f_{\text{NeedleEye}}(x) = \begin{cases} 1 & \textrm{if} \hspace{5pt} \lvert x_i \rvert < eye \hspace{5pt} \forall i \\ \sum_{i=1}^n (100 + \lvert x_i \rvert) & \textrm{if} \hspace{5pt} \lvert x_i \rvert > eye \\ 0 & \textrm{otherwise} \end{cases}$

Where, in this exercise, $eye = 0.0001$ .

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

Two-dimensional NeedleEye function

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

Gavana, A. Global Optimization Benchmarks and AMPGO

class NewFunction01(dimensions=2)¶

NewFunction01 objective function.

This class defines the NewFunction01 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{NewFunction01}}(x) = \left | {\cos\left(\sqrt{\left|{x_{1}^{2} + x_{2}}\right|}\right)} \right |^{0.5} + (x_{1} + x_{2})/100$

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

Two-dimensional NewFunction01 function

Global optimum: $f(x) = -0.18459899925$ for $x = [-8.46669057, -9.99982177]$

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

class NewFunction02(dimensions=2)¶

NewFunction02 objective function.

This class defines the NewFunction02 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{NewFunction02}}(x) = \left | {\sin\left(\sqrt{\lvert{x_{1}^{2} + x_{2}}\rvert}\right)} \right |^{0.5} + (x_{1} + x_{2})/100$

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

Two-dimensional NewFunction02 function

Global optimum: $f(x) = -0.19933159253$ for $x = [-9.94103375, -9.99771235]$

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

Warning

minimum value is estimated from running many optimisations and choosing the best.

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