Griewank function

First order Griewank function

In mathematics, the Griewank function is often used in testing of optimization. It is defined as follows:

[math]\displaystyle{ 1+ \frac {1}{4000} \sum _{i=1}^n x_i^2 -\prod _{i=1}^n \cos \left( \frac {x_i}{\sqrt {i}} \right) }[/math]

The following paragraphs display the special cases of first, second and third order Griewank function, and their plots.

First-order Griewank function

[math]\displaystyle{ g := 1+(1/4000)\cdot x_1^2-\cos(x_1) }[/math]

The first order Griewank function has multiple maxima and minima.

Let the derivative of Griewank function be zero：

[math]\displaystyle{ \frac{1}{2000} \cdot x_1+\sin(x_1) = 0 }[/math]

Find its roots in the interval [−100..100] by means of numerical method,

In the interval [−10000,10000], the Griewank function has 6365 critical points.

Second-order Griewank function

2nd order Griewank function 3D plot

2nd-order Griewank function contour plot

[math]\displaystyle{ 1+\frac {1}{4000} x_1^2 + \frac {1}{4000} x_2^2- \cos(x_1) \cos \left( \frac 1 2 x_2\sqrt {2} \right) }[/math]

Third order Griewank function

Third-order Griewank function Maple animation

[math]\displaystyle{ \left\{ 1+\frac {1}{4000}\,x_1^2 + \frac {1}{4000}\,x_2^2 + \frac {1}{4000}\,{x_{{3}}}^{2}-\cos(x_1) \cos \left( \frac 1 2 x_2 \sqrt {2} \right) \cos \left( \frac 1 3 x_3 \sqrt {3} \right) \right\} }[/math]