Rosenbrock function

In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. 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. The function is defined by

f ( x , y ) = ( a − x ) 2 + b ( y − x 2 ) 2 {\displaystyle f(x,y)=(a-x)^{2}+b(y-x^{2})^{2}}

It has a global minimum at ( x , y ) = ( a , a 2 ) {\displaystyle (x,y)=(a,a^{2})} , where f ( x , y ) = 0 {\displaystyle f(x,y)=0} . Usually, these parameters are set such that a = 1 {\displaystyle a=1} and b = 100 {\displaystyle b=100} . Only in the trivial case where a = 0 {\displaystyle a=0} the function is symmetric and the minimum is at the origin.

Multidimensional generalizations Two variants are commonly encountered.

One is the sum of N / 2 {\displaystyle N/2} uncoupled 2D Rosenbrock problems, and is defined only for even N {\displaystyle N} s:

f ( x ) = f ( x 1 , x 2 , … , x N ) = ∑ i = 1 N / 2 [ 100 ( x 2 i − 1 2 − x 2 i ) 2 + ( x 2 i − 1 − 1 ) 2 ] . {\displaystyle 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}+(x_{2i-1}-1)^{2}\right].}

This variant has predictably simple solutions. A second, more involved variant is

f ( x ) = ∑ i = 1 N − 1 [ 100 ( x i + 1 − x i 2 ) 2 + ( 1 − x i ) 2 ] where x = ( x 1 , … , x N ) ∈ R N . {\displaystyle f(\mathbf {x} )=\sum _{i=1}^{N-1}[100(x_{i+1}-x_{i}^{2})^{2}+(1-x_{i})^{2}]\quad {\mbox{where}}\quad \mathbf {x} =(x_{1},\ldots ,x_{N})\in \mathbb {R} ^{N}.}

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