Rosenbrock function

From Wikipedia, the free encyclopedia

Gradient descent search over the Rosenbrock function.

The Rosenbrock function is a mathematical function used as a test problem for optimisation algorithms. It is also known as Rosenbrock's valley or Rosenbrock's banana function. It is defined by:

f (x, y) = (1 - x) 2 + 100(y - x 2) 2

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:

$f(x) = \sum_{i=1}^{N-1} \left[ (1-x_i)^2+ 100 (x_{i+1} - x_i^2 )^2 \right] \quad \forall x\in\mathbb{R}^N$