Ackley function

In mathematical optimization, the Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed by David Ackley in his 1987 PhD dissertation. The function is commonly used as a minimization function with global minimum value 0 at 0,.., 0 in the form due to Thomas Bäck. While Ackley gives the function as an example of "fine-textured broadly unimodal space" his thesis does not actually use the function as a test.

On an $d$-dimensional domain it is defined as:

$f(𝐱)=-a\exp (-b\sqrt{\frac{1}{d}\sum _{i=1}^d{x}_i^2})$

$-\exp (\frac{1}{d}\sum _{i=1}^d\cos (c{x}_i))+a+\exp (1)$

Recommended variable values are $a=20$, $b=0.2$, and $c=2\pi$.

The global minimum is $f({𝐱}^{\ast })=0$ at ${𝐱}^{\ast }=\mathrm{𝟎}$.

See also

• Test functions for optimization