# Ackley function

Short description: Function used as a performance test problem for optimization algorithms
Ackley function of two variables
Contour surfaces of Ackley's function in 3D

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.[1]

On a 2-dimensional domain it is defined by:

\displaystyle{ \begin{align} f(x,y) = -20&{}\exp\left[-0.2\sqrt{0.5(x^2+y^2)}\,\right] \\ & {} -\exp\left[0.5\left(\cos 2\pi x + \cos 2\pi y \right)\right] + e + 20 \end{align} }

Its global optimum point is

$\displaystyle{ f(0,0) = 0. }$

## Notes

1. Ackley, D. H. (1987) "A connectionist machine for genetic hillclimbing", Kluwer Academic Publishers, Boston MA.