Himmelblau's function

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Himmelblau's function

In 3D
Contour


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. [1]

See also[edit]

References[edit]

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