vector whose $i^{th}$ element $c_i$ is $-{\bm{a}}_i^\top {\bm{a}}_i$, see for example [@Matousek:2007ub] or our Appendix \[app:simplex\_screening\].

\[cor:min\_enclosing\_ball\] For the minimum enclosing ball problem we have the screening rule $$\begin{aligned}
\label{eq:ballcor}
(\bm{e}_i-{\bm{x}})^\top &(2A^\top A{\bm{x}}+{\bm{c}}) > \notag \\ &2\sqrt{\tfrac{1}{2}\max_i({\bm{x}}-\bm{e}_i)^\top (2A^\top A{\bm{x}}+{\bm{c}})}{\left\lVert{\bm{a}}_i-A{\bm{x}}\right\rVert}  \notag
\\ \ \Rightarrow \ x_i^\star = 0 \ .\end{aligned}$$

Our result improves upon the known rules by [@Kallberg:2014kd; @Ahipasaoglu:2008il] by providing a broader selection criterion .

$L_1$-Constrained Problems {#subsec:l1ball_screening}
--------------------------

$L_1$-constrained formulations are very widely used in order to induce sparsity in the variables. Here below we provide results for screening on general $L_1$-constrained problems, that is $\min_{{\bm{x}}\in{\mathcal{C}}} f(A{\bm{x}})$ for ${\mathcal{C}}= L_1 \subset{\mathbb{R}}^n$ (or a scaled version of the $L_1$-ball). Proofs are provided in Appendix \[app:l1ball\_screening\].

\[thm:l1\_constrained\_screening\] For general $L_1$-constrained optimization $\min_{{\bm{x}}\in L_1} f(A{\bm{x}})$, the optimality condition  gives rise to the following screening rule at the optimal point, for any $i \in [n]$ $$\label{eq:l1_constrained}
{\left\lvert{\bm{a}}_i^\top {\bm{w}}^\star\right\rvert}  + (A {\bm{x}}^\star)^\top{\bm{w}}^\star <0 
\ \Rightarrow \ x_i^\star  = 0\ .$$

Using only a current iterate ${\bm{x}}$ instead of an optimal point, we obtain screening for general smooth and strongly convex function $f$:

\[thm:l1\_smooth\_constrained\] Let $f$ be $L$-smooth and $\mu$-strongly convex over the $L_1$-ball. Then for $L_1$-constrained optimization $\min_{{\bm{x}}\in L_1} f(A{\bm{x}})$ we have the 