following screening rule, for any $i \in [n]$ $$\begin{aligned}
{\left\lvert{\bm{a}}_i^\top \nabla f(A{\bm{x}})\right\rvert} + &(A{\bm{x}})^\top \nabla f(A{\bm{x}}) \notag \\ +& L({\left\lVert{\bm{a}}_i\right\rVert}_2 + {\left\lVertA{\bm{x}}\right\rVert}_2)\sqrt{\frac{{G_{\mathcal{C}}}({\bm{x}})}{\mu}} < 0  \notag
\\ \Rightarrow \ {\bm{x}}_i^\star = 0 \label{eq:l1_constrained_screening}\end{aligned}$$

Elastic Net Constrained Problems
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Elastic net regularization as an alternative to $L_1$ is often used in practice, and can outperform the Lasso, while still enjoying a similar sparsity of representation [@Zou:2005EN]. The elastic net is given by the expression $$\alpha {\left\lVert{\bm{x}}\right\rVert}_1 + \frac{(1 - \alpha) }{2} {\left\lVert{\bm{x}}\right\rVert}_2^2
\ .$$ Here below we provide novel result for screening on general elastic net constrained problems, that is $\min_{{\bm{x}}\in{\mathcal{C}}} f(A{\bm{x}})$ for ${\mathcal{C}}$ being the elastic net constraint, or a scaled version of it. Proofs are provided in Appendix \[app:elastic\_net\_screening\].

\[thm:elastic\_net\_constrained\] For general elastic net constrained optimization $\min_{{\bm{x}}\in L_E} f(A{\bm{x}})$ where $L_E := \{{\bm{x}}\in{\mathbb{R}}^n\ |\ \alpha {\left\lVert{\bm{x}}\right\rVert}_1 + \frac{(1 - \alpha) }{2} {\left\lVert{\bm{x}}\right\rVert}_2^2 \le 1\}$, the optimality condition  gives rise to the following screening rule at the optimal point, for any $i \in [n]$ $$\begin{aligned}
|{\bm{a}}_i^\top {\bm{w}}^\star | + (A{\bm{x}}^\star)^\top  {\bm{w}}^\star \Big [ \frac{ \alpha  }{ 1 + \frac{(1 - \alpha) }{2} \|{\bm{x}}^\star\|_2^2} \Big ] < 0  \Rightarrow x_i^\star = 0\end{aligned}$$

Using only a current iterate ${\bm{x}}$ instead of an optimal 