Himmelblau's function

{{short description|Function used as a performance test problem for optimization algorithms}}

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

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

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

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

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|date=November 2011}}

The function is named after David Mautner Himmelblau (1924–2011), who introduced it.{{cite book |last=Himmelblau |first=D. |title=Applied Nonlinear Programming |publisher=McGraw-Hill |year=1972 |isbn=0-07-028921-2 }}