Rosenbrock function

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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.


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