point, we obtain screening for general smooth and strongly convex function $f$:

\[thm:elastic\_smooth\_constrained\] Let $f$ be $L$-smooth and $\mu$-strongly convex over the elastic net norm ball. Then for elastic net constrained optimization $\min_{{\bm{x}}\in L_E} f(A{\bm{x}})$ we have the following screening rule, for any $i \in [n]$ $$\begin{aligned}
{\left\lvert{\bm{a}}_i^\top \nabla f(A{\bm{x}})\right\rvert} + &(A{\bm{x}})^\top \nabla f(A{\bm{x}}) \big[ \frac{2\alpha}{3-\alpha} \big ] \notag \\ +& L({\left\lVert{\bm{a}}_i\right\rVert}_2 + {\left\lVertA{\bm{x}}\right\rVert}_2\big[ \frac{2\alpha}{3-\alpha} \big ])\sqrt{\frac{{G_{\mathcal{C}}}({\bm{x}})}{\mu}} < 0  \notag
\\ \Rightarrow \ {\bm{x}}_i^\star = 0 \label{eq:l1_constrained_screening}\end{aligned}$$

Note that both above results also recover the $L_1$ constrained case as a special case, when $\alpha \rightarrow 1$.

Screening for Box Constrained Problems {#subsec:box_screening}
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Box-constrained problems are important in several machine learning applications, including SVMs. After variable rescaling, w.l.o.g. we can assume the constraint set ${\mathcal{C}}= \Box := \{{\bm{x}}\in{\mathbb{R}}^n \ | \ 0 \le x_i \le 1 \}$. We derive screening rules for predicting both if a variable will take the upper or lower constraint.

\[thm:box\_constrained\] Let $f$ be $L$-smooth. Then for box-constrained optimization $\min_{{\bm{x}}\in \Box} f(A{\bm{x}})$, we obtain the following screening rules, for any $i \in [n]$ $$\begin{aligned}
&{\bm{a}}_i^\top \nabla f(A{\bm{x}}) -  {\left\lVert{\bm{a}}_i\right\rVert}_2\sqrt{2L {G}({\bm{x}})}>0 
\ \Rightarrow \  x_i^\star =0 \ , \ \text{ and}\\
&{\bm{a}}_i^\top \nabla f(A{\bm{x}}) +  {\left\lVert{\bm{a}}_i\right\rVert}_2\sqrt{2L\,{G}({\bm{x}})} < 