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Ackley function

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Function used as a performance test problem for optimization algorithms Ackley function of two variablesContour 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. 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} .

See also

Notes

  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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