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Griewank Function


GriewankFunction

The Griewank function is a function widely used to test the convergence of optimization functions. The Griewank function of order n is defined by

 f_n(x_1,...,x_n)=1+1/(4000)sum_(i=1)^nx_i^2-product_(i=1)^ncos((x_i)/(sqrt(i)))

for x_i in [-600,600] (Griewank 1981), plotted above for n=1. It has a global minimum of 0 at the point x=0.

GriewankFunctionZeros

The function f_1(x) has 191 minima, with global minimum at x=0 and local minima at +/-x for x approx 6.28005 (OEIS A177889), 12.5601, 18.8401, 25.1202, .... Restricting the domain of the function to [-k,k], the numbers of local minima for f_1^((k)) for k=1, 2, ... are therefore given by 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, ... (OEIS A178832).


See also

Rosenbrock Function

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References

Cho, H.; Olivera, F.; and Guikema, S. D. "A Derivation of the Number of Minima of the Griewank Function." Appl. Math. Comput. 204, 694-701, 2008.Griewank, A. O. "Generalized Decent for Global Optimization." J. Opt. Th. Appl. 34, 11-39, 1981.Locatelli, M. "A Note on the Griewank Test Function." J. Global Opt. 25, 169-174, 2003.Sloane, N. J. A. Sequences A177889 and A178832 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Griewank Function

Cite this as:

Weisstein, Eric W. "Griewank Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GriewankFunction.html

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