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:

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

Himmelblau's function

In 3D

Log-spaced level curve plot

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

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

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.^{[citation needed]}

The function is named after David Mautner Himmelblau (1924–2011), who introduced it.^[1]

See also

• Test functions for optimization