# Rosenbrock function

The original form has 2 variables:

$LaTeX: f(x_1, x_2) = 100(x_2 - x_{1}^{2})^2 + (1 - x_1)^2$

The significance of this function is that it has "bannana-shaped" contours, making it difficult for nonlinear programming algorithms, particularly Cauchy's steepest descent, to solve it. The (global) minimum is at $LaTeX: \textstyle x^* = (1, 1)$ with $LaTeX: \textstyle f(x^*) = 0.$ There are variations that extend Rosenbrock's function, many by Schittkowski, such as the following on $LaTeX: \mathbb{R}^n:$

$LaTeX: f(x) = \sum_{1 \le j \le n/2} 100(x_{2j} - x_{2j-1}^2)^2 + (1 - x_{2j-1})^2.$