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Griewank function

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Griewank function

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

Contents

1 + 1 4000 ∑ i = 1 n x i 2 − ∏ i = 1 n cos ⁡ ( x i i )

The following paragraphs display the special cases of first,second and third order Griewank function, and their plots.

First-order Griewank function

g := 1 + ( 1 / 4000 ) ⋅ x [ 1 ] 2 − cos ⁡ ( x [ 1 ] )

First order Griewank function has multiple maxima and minima.

Let the derivative of Griewank function be zero：

1 2000 ⋅ x [ 1 ] + sin ⁡ ( x [ 1 ] ) = 0

Find its roots in the interval [−100..100] by means of numerical method,

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

Second-order Griewank function

1 + 1 4000 x 1 2 + 1 4000 x 2 2 − cos ⁡ ( x 1 ) cos ⁡ ( 1 2 x 2 2 )

Third order Griewank function

{ 1 + 1 4000 x 1 2 + 1 4000 x 2 2 + 1 4000 x 3 2 − cos ⁡ ( x 1 ) cos ⁡ ( 1 2 x 2 2 ) cos ⁡ ( 1 3 x 3 3 ) }

References

Griewank function Wikipedia
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