Ackley function

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:

{\begin{aligned}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{aligned}}

^[2]

Its global optimum point is

f(0,0)=0.

Notes

^ Ackley, D. H. (1987) "A connectionist machine for genetic hillclimbing", Kluwer Academic Publishers, Boston MA.
^ Bäck, Thomas (1996-02-15). "Artificial Landscapes". Evolutionary Algorithms in Theory and Practice. Oxford University Press. p. 142. doi:10.1093/oso/9780195099713.003.0008. ISBN 978-0-19-509971-3.