Ackley function

{{short description|Function used as a performance test problem for optimization algorithms}}

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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.Ackley, D. H. (1987) "[https://books.google.com/books?id=sx_VBwAAQBAJ&q=%22Ackley+function%22 A connectionist machine for genetic hillclimbing]", Kluwer Academic Publishers, Boston MA. p. 13-14 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 $d$ dimensions, is defined as{{Cite web |url=https://www.sfu.ca/~ssurjano/ackley.html |title=Ackley Function |last=Bingham |first=Derek |date=2013 |website=Virtual Library of Simulation Experiments: Test Functions and Datasets |publisher=Simon Fraser University |access-date=December 22, 2024}}

: $$

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(c x_i) \right) + a + \exp(1)

Recommended variable values are $a\; =\; 20$, $b\; =\; 0.2$, and $c\; =\; 2\backslash pi$.

The global minimum is $f(x^*)\; =\; 0$ at $x^*\; =\; 0$.