Rosenbrock function

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Plot of Rosenbrock function of two variables.

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:

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

For $N \ge 4$ , the function has at least one local minimum in the neighborhood of $(x_1, x_2, \dots, x_N) = (-1, 1, \dots, 1)$ in addition to the trivial global minimum at $(x_1, \dots, x_N) = (1, \dots, 1)$ ^[1].