Griewank function

First order Griewank function

In mathematics, the Griewank function is often used in testing of optimization, and is defined as follows;

$1+{\frac {1}{4000}}\sum _{i=1}^{n}x_{i}^{2}-\prod _{i=1}^{n}\cos \left({\frac {x_{i}}{\sqrt {i}}}\right)$

First-order Griewank function

$g:=1+(1/4000)\cdot x[1]^{2}-\cos(x[1])$

First order Griewank function has multiple maxima and minima.

Let the derivative of Griewank function be zero：

${\frac {1}{2000}}\cdot x[1]+\sin(x[1])=0$

Find its roots in the interval [−100..100] by means of numerical method, and obtain 62 solutions：

[−97.438110610025603200, −94.200661844477748520, −91.151778636270389965, −87.920619819329359985, −84.865447114417916660, −81.640577359698817225, −78.579116013127494725, −75.360534496781834905, −72.292785301096032350, −69.080491261738200370, −66.006454947055212230, −62.800447685694571355, −59.720124919768677970, −56.520403799747265470, −53.433795188029228405, −50.240359634965042195, −47.147465720656019171, −43.960315222391878044, −40.861136486491770843, −37.680270593049735600, −34.574807454399982858, −31.400225777941327138, −28.288478593262152626, −25.120180808052873462, −22.002149871974999083, −18.840135714356858698, −15.715821259447690012, −12.560090527814781691, −9.4294927245990724370, −6.2800452793799046870, −3.1431642363549054240, 3.1431642363549054240, 6.2800452793799046870, 9.4294927245990724370, 12.560090527814781691, 15.715821259447690012, 18.840135714356858698, 22.002149871974999083, 25.120180808052873462, 28.288478593262152626, 31.400225777941327138, 34.574807454399982858, 37.680270593049735600, 40.861136486491770847, 43.960315222391878044, 47.147465720656019171, 50.240359634965042195, 53.433795188029228405, 56.520403799747265470, 59.720124919768677970, 62.800447685694571355, 66.006454947055212230, 69.080491261738200370, 72.292785301096032350, 75.360534496781834905, 78.579116013127494725, 81.640577359698817225, 84.865447114417916660, 87.920619819329359985, 91.151778636270389965, 94.200661844477748520, 97.438110610025603200, 0.]

In the interval [−10000,10000], the Griewank function has 6365 critical points.

Second-order Griewank function

2nd order Griewank function 3D plot

2nd-order Griewank function contour plot

$1+{\frac {1}{4000}}x_{1}^{2}+{\frac {1}{4000}}x_{2}^{2}-\cos(x_{1})\cos \left({\frac {1}{2}}x_{2}{\sqrt {2}}\right)$

Third order Griewank function

Third-order Griewank function Maple animation

$\left\{1+{\frac {1}{4000}}\,x_{1}^{2}+{\frac {1}{4000}}\,x_{2}^{2}+{\frac {1}{4000}}\,{x_{3}}^{2}-\cos(x_{1})\cos \left({\frac {1}{2}}x_{2}{\sqrt {2}}\right)\cos \left({\frac {1}{3}}x_{3}{\sqrt {3}}\right)\right\}$