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\begin{itemize}  \item \textit{The Rosenbrock function:} We define the Rosenbrock function mapping $\mathbf{R}^2$ to $\mathbf{R}$ as $f(x,y) = 100 \cdot (y - x^2)^2 + (1 - x)^2$. The function has a global minimum at $(0,0)$ that lies within a a long, flat valley. We define the function and its gradient as R object, and then run the optimization routine:  \begin{lstlisting}  objective <- function(x) {   100 * (x[2] - x[1]^2)^2 + (1 - x[1])^2  }  gradient <- function(x) {  c(-400 * x[1] * (x[2] - x[1]^2) - 2 * (1 - x[1]),  200 * (x[2] - x[1]^2))  }  out <- lbfgs(objective, gradient, c(-1.2, 1))  \end{lstlisting}  \end{itemize}