The Griewank function is a function widely used to test the convergence of optimization algorithms. The Griewank function of order , for a positive
integer, is defined by
for
(Griewank 1981), plotted above for . It has a unique global minimum
of 0 at the point .
The function
has 191 interior local minima on , with global minimum
at
and local minima at
for
(OEIS A177889), 12.5601, 18.8401, 25.1202,
.... Restricting the domain of the function to , the numbers of interior local minima for , 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).
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 Descent for Global
Optimization." J. Opt. Th. Appl.34, 11-39, 1981. https://doi.org/10.1007/BF00933356.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."
, for a positive
integer 
, is defined by
![x_i in [-600,600]](/images/equations/GriewankFunction/Inline3.svg)
(Griewank 1981), plotted above for 
. It has a unique global minimum
of 0 at the point 
.
has 191 interior local minima on ![[-600,600]](/images/equations/GriewankFunction/Inline7.svg)
, with global minimum
at 
and local minima at 
for 
(OEIS A177889), 12.5601, 18.8401, 25.1202,
.... Restricting the domain of the function to ![[-k,k]](/images/equations/GriewankFunction/Inline11.svg)
, the numbers of interior local minima for 
, 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).
