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| [[Image:Rosenbrock function.svg|thumb|right|300 px|Plot of the Rosenbrock function of two variables.]]
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| In [[mathematical optimization]], the '''Rosenbrock function''' is a non-[[convex function]] used as a performance test problem for optimization [[algorithm]]s introduced by [[Howard Harry Rosenbrock|Howard H. Rosenbrock]] in 1960.<ref>{{cite journal|last=Rosenbrock|first=H.H.|title=An automatic method for finding the greatest or least value of a function|journal=The Computer Journal|year=1960|volume=3|pages=175–184|doi=10.1093/comjnl/3.3.175| issn=0010-4620 }}</ref> It is also known as '''Rosenbrock's valley''' or '''Rosenbrock's banana function'''.
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| The global minimum is inside a long, narrow, [[parabolic]] shaped flat valley. To find the valley is trivial. To converge to the global [[minimum]], however, is difficult.
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| The function is defined by
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| : <math>f(x, y) = (1-x)^2 + 100(y-x^2)^2 .\quad </math>
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| It has a global minimum at <math>(x, y)=(1, 1)</math> where <math>f(x, y)=0</math>. A different coefficient of the second term is sometimes given, but this does not affect the position of the global minimum.
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| ==Multidimensional generalisations==
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| Two variants are commonly encountered. One is the sum of <math>N/2</math> uncoupled 2D Rosenbrock problems,
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| : <math>f(\mathbf{x}) = f(x_1, x_2, \dots, x_N) = \sum_{i=1}^{N/2} \left[100(x_{2i-1}^2 - x_{2i})^2
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| + (x_{2i-1} - 1)^2 \right].</math><ref>L C W Dixon, D J Mills. Effect of Rounding errors on the Variable Metric Method. ''Journal of Optimization Theory and Applications'' '''80''', 1994. [http://portal.acm.org/citation.cfm?id=179711]</ref>
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| This variant is only defined for even <math>N</math> and has predictably simple solutions.
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| A more involved variant is
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| : <math>f(\mathbf{x}) = \sum_{i=1}^{N-1} \left[ (1-x_i)^2+ 100 (x_{i+1} - x_i^2 )^2 \right] \quad \forall x\in\mathbb{R}^N.</math><ref>{{cite web |url= http://www.it.lut.fi/ip/evo/functions/node5.html |title=Generalized Rosenbrock's function |accessdate=2008-09-16 |work= |publisher= |date= }}</ref>
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| This variant has been shown to have exactly one minimum for <math>N=3</math> (at <math>(1, 1, 1)</math>) and exactly two minima for <math>4 \le N \le 7</math> -- the global minimum of all ones and a local minimum near <math>(x_1, x_2, \dots, x_N) = (-1, 1, \dots, 1)</math>. This result is obtained by setting the gradient of the function equal to zero, noticing that the resulting equation is a rational function of <math>x</math>. For small <math>N</math> the polynomials can be determined exactly and [[Sturm's theorem]] can be used to determine the number of real roots, while the roots can be [[Fundamental theorem of algebra#Bounds on the zeroes of a polynomial|bounded]] in the region of <math>|x_i| < 2.4</math>.<ref name="kok2009">Schalk Kok, Carl Sandrock. Locating and Characterizing the Stationary Points of the Extended Rosenbrock Function. ''Evolutionary Computation'' '''17''', 2009. [http://www.mitpressjournals.org/doi/abs/10.1162/evco.2009.17.3.437]</ref> For larger <math>N</math> this method breaks down due to the size of the coefficients involved.
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| ==Stationary points==
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| Many of the stationary points of the function exhibit a regular pattern when plotted.<ref name="kok2009"/> This structure can be exploited to locate them.
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| [[Image:Rosenbrock roots exhibiting hump structures.pdf|thumb|right|300 px|Rosenbrock roots exhibiting hump structures]]
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| == An example of optimization ==
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| The Rosenbrock function can be efficiently optimized by adapting appropriate coordinate system without using any [[gradient descent|gradient information]] and without building local approximation models (in contrast to many derivate-free optimizers). The following figure illustrates an example of 2-dimensional Rosenbrock function optimization by
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| [[Adaptive coordinate descent]] from starting point <math>x_0=(-3,-4)</math>. The solution with the function value <math>10^{-10}</math> can be found after 325 function evaluations.
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| [[File:Rosenbrock.png|thumb|right|x300px]] | |
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| == See also ==
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| *[[Test functions for optimization]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{Citation | last1=Rosenbrock | first1=H. H. | title=An automatic method for finding the greatest or least value of a function | doi=10.1093/comjnl/3.3.175 | id={{MathSciNet | id = 0136042}} | year=1960 | journal=The Computer Journal | issn=0010-4620 | volume=3 | pages=175–184}}
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| ==External links==
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| * [http://www.gnuplot.info/screenshots/figs/pm3d-Rosenbrock.png Rosenbrock function plot in 3D]
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| * [http://demonstrations.wolfram.com/MinimizingTheRosenbrockFunction/ Minimizing the Rosenbrock Function] by Michael Croucher, [[The Wolfram Demonstrations Project]].
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| * {{MathWorld |title=Rosenbrock Function |urlname=RosenbrockFunction}}
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| [[Category:Mathematical optimization]]
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The writer's name is Christy. He is an order clerk and it's some thing he truly appreciate. Kentucky is exactly where I've always been living. To climb is some thing I truly enjoy doing.
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