Rosenbrock function

In mathematical optimization, the Rosenbrock function is a non-convex function used as a test problem for optimization algorithms. It is also known as Rosenbrock's valley or Rosenbrock's banana function.

This function is often used to test performance of optimization algorithms. The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial, however to converge to the global minimum is difficult.

It is defined by

: $f(x,\; y)\; =\; (1-x)^2\; +\; 100(y-x^2)^2\; .quad$

It has a global minimum at $(x,\; y)=(1,\; 1)$ where $f(x,\; y)=0$. A different coefficient of the second term is sometimes given, but this does not affect the position of the global minimum.

A common multidimensional extension is

: $f(x)\; =\; sum\_\{i=1\}^\{N-1\}\; left\; [\; (1-x\_i)^2+\; 100\; (x\_\{i+1\}\; -\; x\_i^2\; )^2\; ight]\; quad\; forall\; xinmathbb\{R\}^N.$ [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= ]

For $N\; ge\; 4$, the function has at least one local minimum in the neighborhood of $(x\_1,\; x\_2,\; dots,\; x\_N)\; =\; (-1,\; 1,\; dots,\; 1)$ inaddition to the trivial global minimum at$(x\_1,\; dots,\; x\_N)\; =\; (1,\; dots,\; 1).$ [Yun-Wei Shang, Yu-Huang Qiu.A Note on the Extended Rosenbrock Function."Evolutionary Computation" 14, 2006. [http://www.mitpressjournals.org/doi/abs/10.1162/evco.2006.14.1.119] ]

Notes

External links

* [http://www.gnuplot.info/screenshots/figs/pm3d-Rosenbrock.pngRosenbrock function plot in 3D]
* [http://www.rpi.edu/~vanfrl/rosenbrock_minim.gifMinimization of Rosenbrock's function] using the BFGS, DFP, and Steepest Descent Algorithms 2D plot
* [http://demonstrations.wolfram.com/MinimizingTheRosenbrockFunction/ Minimizing the Rosenbrock Function] by Michael Croucher, The Wolfram Demonstrations Project.
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