screening in hinge loss SVM is given in [@Zimmert:2015ue] and [@Shibagaki:2016ts]. The work of [@Zimmert:2015ue] also covers the kernelized SVM case, and improves the threshold given in our Corollary \[cor:SVMhinge\] by a constant of $\sqrt{2}$. In Appendix \[app:box\_screening\], we show that our more general approach here can also be adjusted to gain this constant factor.

Screening for Penalized Problems {#sec:screening_penalized}
================================

In this section we will develop screening methods for general penalized convex optimization problems of the form and . The cornerstone application are $L_1$ regularized problems, for which we now develop screening rules with general cost function $f$. We show in Appendix \[app:penalized\_problem\_l1\] that our method can reproduce the screening rules of [@Ndiaye:2015wj] as special cases, whereas their method does not directly extend to general $f$. Beyond $L_1$ problems, we also describe new screening rules for elastic net regularized problems, as well as the important case of structured norm regularized optimization.

$L_1$-Penalized Problems
------------------------

The next theorem describes a screening rule for general $L_1$-penalized problems, under a smoothness assumption on function $f$. Proofs for are given in Appendix \[app:penalized\_problem\_l1\].

\[thm:smooth\_pen\_lasso\] Consider an $L_1$-regularized optimization problem of the form $$\begin{aligned}
\label{eq:l1_regularized}
 \min_{{\bm{x}}\in {\mathbb{R}}^n} \ \ f(A{\bm{x}}) +  \lambda {\left\lVert{\bm{x}}\right\rVert}_1\end{aligned}$$ If 