Rosenbrock function
Encyclopedia
In mathematical optimization, the Rosenbrock function is a non-convex function
used as a performance test problem for optimization algorithm
s introduced by Howard H. Rosenbrock
in 1960. It is also known as Rosenbrock's valley or Rosenbrock's banana function.
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.
It is defined by
It has a global minimum at
where 
. A different coefficient of the second term is sometimes given, but this does not affect the position of the global minimum.
uncoupled 2D Rosenbrock problems,
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
used as a performance test problem for optimization algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s introduced by Howard H. Rosenbrock
Howard Harry Rosenbrock
Howard Harry Rosenbrock was a leading figure in control theory and control engineering. He was born in Ilford, England in 1920, graduated in 1941 from University College London with a 1st class honors degree in Electrical Engineering. He served in the Royal Air Force during World War II. He...
in 1960. It is also known as Rosenbrock's valley or Rosenbrock's banana function.
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.
It is defined by
It has a global minimum at
where . A different coefficient of the second term is sometimes given, but this does not affect the position of the global minimum.Multidimensional generalisations
Two variants are commonly encountered. One is the sum of uncoupled 2D Rosenbrock problems,
-
This variant is only defined for even
and has predictably simple solutions.
A more involved variant is
This variant has been shown to have exactly one minimum for
(at 
) and exactly two minima for 
-- the global minimum of all ones and a local minimum near 
. This result is obtained by setting the gradient of the function equal to zero, noticing that the resulting equation is a rational function of 
. For small 
the polynomials can be determined exactly and Sturm's theoremSturm's theoremIn mathematics, Sturm's theorem is a symbolic procedure to determine the number of distinct real roots of a polynomial. It was named for Jacques Charles François Sturm...
can be used to determine the number of real roots, while the roots can be bounded in the region of
. For larger 
this method breaks down due to the size of the coefficients involved.
Stationary points
Many of the stationary points of the function exhibit a regular pattern when plotted. This structure can be exploited to locate them.
Stochastic version
There are many ways to extend this function stochastically. In Xin-She Yang's functions, a
generic or heuristic extension of Rosenbrock's function is given to extend this function into a stochasticStochasticStochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
function
where the random variables
obey
a uniform distribution Unif(0,1). In principle, this stochastic
function has the same global optimimum at (1,1,...,1), however, the stochastic nature
of this function makes it impossible to use any gradient-basedGradient descentGradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient of the function at the current point...
optimizationOptimizationOptimization or optimality may refer to:* Mathematical optimization, the theory and computation of extrema or stationary points of functionsEconomics and business* Optimality, in economics; see utility and economic efficiency...
algorithms.
External links
- Rosenbrock function plot in 3D
- Minimizing the Rosenbrock Function by Michael Croucher, The Wolfram Demonstrations Project.
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