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] 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:^[2]

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

$- \exp (\frac{1}{d} \sum_{i = 1}^{d} \cos (c x_{i})) + a + \exp (1)$

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

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

Notes

↑ Ackley, D. H. (1987) "A connectionist machine for genetic hillclimbing", Kluwer Academic Publishers, Boston MA. p. 13-14
↑ Bingham, Derek (2013). "Ackley Function". Virtual Library of Simulation Experiments: Test Functions and Datasets. Simon Fraser University. Retrieved December 22, 2024.

[1] Ackley, D. H. (1987) "A connectionist machine for genetic hillclimbing", Kluwer Academic Publishers, Boston MA. p. 13-14

[2] Bingham, Derek (2013). "Ackley Function". Virtual Library of Simulation Experiments: Test Functions and Datasets. Simon Fraser University. Retrieved December 22, 2024.

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See also

Notes