Griewank function

In mathematical optimization, the Griewank function is used as a performance test problem for optimization algorithms. For order $n$ it is defined by:

$f(x_{1},x_{2},\ldots ,x_{n}):=1+{\frac {1}{4000}}\sum _{i=1}^{n}x_{i}^{2}-\prod _{i=1}^{n}\cos \left({\frac {x_{i}}{\sqrt {i}}}\right).$

The Griewank function has a global minimum at $f(0,0,\ldots ,0)=0$. A typical search area is $x_{i}=[-600,600]$, for $i=1,\ldots ,n$ .

First-order Griewank function

$g:=1+(1/4000)\cdot x[1]^{2}-\cos(x[1])$

First order Griewank function has multiple maxima and minima.

Let the derivative of Griewank function be zero：

${\frac {1}{2000}}\cdot x[1]+\sin(x[1])=0$

Second-order Griewank function

$1+{\frac {1}{4000}}x_{1}^{2}+{\frac {1}{4000}}x_{2}^{2}-\cos(x_{1})\cos \left({\frac {1}{2}}x_{2}{\sqrt {2}}\right)$

Third order Griewank function

$\left\{1+{\frac {1}{4000}}\,x_{1}^{2}+{\frac {1}{4000}}\,x_{2}^{2}+{\frac {1}{4000}}\,{x_{3}}^{2}-\cos(x_{1})\cos \left({\frac {1}{2}}x_{2}{\sqrt {2}}\right)\cos \left({\frac {1}{3}}x_{3}{\sqrt {3}}\right)\right\}$

See also

• Test functions for optimization