&\quad  \qquad \lambda_1 - \sqrt{2({\bm{a}}_i^\top {\bm{a}}_i + 2\lambda_2) {G}({\bm{x}})} \ \Rightarrow \ x_i^\star = 0.\end{aligned}$$

We also recover existing screening rules for elastic net regularized problem with more general objective $f$ using our frameworks, in Appendix \[app:penalized\_problem\_l1\], see Lemma \[lem:smooth\_elastic\] and Theorem \[thm:elasticnet\_genral\] which has been earlier derived in [@Shibagaki:2016ts]. In the proof, we derive screening rules from both the formulation and using optimality condition and which is novel as well as help us to understand the property useful in deriving screening rules for elastic net penalized problems.

Structured Norm Penalized Problems {#subsec:l1_l2_genral}
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Here in this section we present screening rules for non-overlapping group norm regularized problems. Group-norm regularization is widely used to induce sparsity in terms of groups of variables of the the solution of the optimization problem. The most prominent example is the group lasso ($\ell_2/\ell_1$-regularization). Here in this section we mostly discuss screening for general objectives with an $\ell_2/\ell_1$-regularization. Proofs are provided in Appendix \[app:structured\_norm\].

#### Group Norm - $\ell_2/\ell_1$ Regularization. {#par:l1_l2_group}

In the following, we use the notation $\{ {\bm{x}}_1 \cdots {\bm{x}}_G\}$ to express a vector ${\bm{x}}$ as a partition of the groups of variables, such that ${\bm{x}}^\top = \left[ {\bm{x}}_1^\top, {\bm{x}}_2^\top \cdots {\bm{x}}_G^\top \right]$. 