Griewank function

First order Griewank function

Griewank function is often used in testing of optimization, it is defined as follow

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

First order Griewank function

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

First order Griewank function has multiple maximas and minimas . let the derivative of Griewank function be zero：

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

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[-10000,10000]interval, Griewank function has =6365 critical points.

Second order Griewank function

2nd order Griewank function 3D plot

2nd order Griewank function contour plot

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

Third order Griewank function

Third order Griewank function Maple animation

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