Himmelblau's function

In mathematical optimization, Himmelblau's function is a multi-modal function, used to test the performance of optimization algorithms. The function is defined by:

$f(x,y)=(x^2+y-11{)}^2+(x+y^2-7{)}^2.$

It has one local maximum at $x=-0.270845$ and $y=-0.923039$ where $f(x,y)=181.617$, and four identical local minima:

• $f(3.0,2.0)=0.0,$
• $f(-2.805118,3.131312)=0.0,$
• $f(-3.779310,-3.283186)=0.0,$
• $f(3.584428,-1.848126)=\mathrm{0.0.}$

The locations of all the minima can be found analytically. However, because they are roots of quartic polynomials, when written in terms of radicals, the expressions are somewhat complicated.Template:Citation needed

The function is named after David Mautner Himmelblau (1924–2011), who introduced it.

See also

• Test functions for optimization

References

Template:Mathanalysis-stub