Himmelblau's function

Function used as a performance test problem for optimization algorithms
Himmelblau's function
In 3D
Log-spaced level curve plot

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

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

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

References

  1. ^ Himmelblau, D. (1972). Applied Nonlinear Programming. McGraw-Hill. ISBN 0-07-028921-2.


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