# Himmelblau's function

Himmelblau's function
In 3D
Log-spaced down level curve plot [1]

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

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

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

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

The locations of all the minima can be found analytically. However, because they are roots of cubic 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.[2]

## References

1. ^ Simionescu, P.A. (2011). "Some Advancements to Visualizing Constrained Functions and Inequalities of Two Variables". Transactions of the ASME - Journal of Computing and Information Science in Engineering 11 (1). doi:10.1115/1.3570770.
2. ^ Himmelblau, D. (1972). Applied Nonlinear Programming. McGraw-Hill. ISBN 0-07-028921-2.