Himmelblau's function

Himmelblau's function

In 3D

Contour

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

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

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

• $f(3.0, 2.0) = 0.0 \quad$,

• $f(-2.805118, 3.131312) = 0.0 \quad$,

• $f(-3.779310, -3.283186) = 0.0 \quad$,

• $f(3.584428, -1.848126) = 0.0 \quad$.

The locations of all the minima can be found analytically, but the expressions are long and complicated.^{[citation needed]}

See also

• Rosenbrock function