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\subsection{Notation}  Throughout this vignette, we will be using the following notation. notation, which largely follows Andrew and Gao (2007).  Let $f: \mathbb{R}^\mathbb{n} \mapsto \mathbb{R}$ be the objective function to be minimized. We also let the $\lvert \lvert \cdot \lvert \lvert$ operator denote the $L_2$ norm of a vector, and $\lvert \lvert \cdot \lvert \lvert_1$ denote the $L_1$ norm. $\mathbf{B_k}$ is the Hessian matrix (or its approximation) of $f$ at $x_k$, and $g_k$ if the gradient of $f$ at the same point.  \subsection{The L-BFGS Algorithm}