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.^[1] 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

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

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