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] 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 ${\displaystyle d}$-dimensional domain it is defined as^[2]:

${\displaystyle f(\mathbf {x} )=-a\exp \left(-b{\sqrt {{\frac {1}{d}}\sum _{i=1}^{d}x_{i}^{2}}}\right)}$

${\displaystyle -\exp \left({\frac {1}{d}}\sum _{i=1}^{d}\cos(cx_{i})\right)+a+\exp(1)}$

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

The global minimum is ${\displaystyle f(\mathbf {x} ^{*})=0}$ at ${\displaystyle \mathbf {x} ^{*}=\mathbf {0} }$.

• Test functions for optimization