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.

For d {\displaystyle d} dimensions, is defined as

f ( x ) = − a exp ⁡ ( − b 1 d ∑ i = 1 d x i 2 ) − exp ⁡ ( 1 d ∑ i = 1 d cos ⁡ ( c x i ) ) + a + exp ⁡ ( 1 ) {\displaystyle f(x)=-a\exp \left(-b{\sqrt {{\frac {1}{d}}\sum _{i=1}^{d}x_{i}^{2}}}\right)-\exp \left({\frac {1}{d}}\sum _{i=1}^{d}\cos(cx_{i})\right)+a+\exp(1)}

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

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