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\subsection{The OWL-QN Algorithm}  The L-BFGS method cannot be applied to problems involving $L_1$ norms with an objective function  of the parameter vector, form $r(x) = C \cdot \lvert \lvert x \lvert \lvert _1 C = \cdor \sum_i \lvert x_i \lvert$,  such as LASSO regression or $L_1$-penalized log-linear models, given the non-differentiability of the objective function at any point where at least one of the parameters is zero. The OWL-QN algorithm exploits the fact that $L_1$-regularized objective functions will stillTrying out the math:  \[  r(x) = C \cdot \lvert \lvert x\lvert \lvert _1 = C \cdot \sum_i \lvert x_i \lvert  \]