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As we have mentioned, any type of screening will crucially rely on first deriving safe knowledge about the unknown optimal points of our given optimization problem. Here, we will use the duality gap to obtain such knowledge on the optimal points ${\bm{x}}^\star\in{\mathbb{R}}^n$ and ${\bm{w}}^\star\in{\mathbb{R}}^d$ of the respective optimization problems and respectively. Proofs are provided in Appendix \[app:optimal\_point\].

Our first lemma shows how to bound the distance between any (feasible) current dual iterate and the solution ${\bm{w}}^\star$ using standard assumptions on the objective functions.

\[lem:stronglyconvex\] Consider the problem with optimal solution ${\bm{w}}^\star\in {\mathbb{R}}^d$. For $f$ being $\mu$-smooth, we have $${\left\lVert{\bm{w}}- {\bm{w}}^\star\right\rVert}^2 \le \frac{2}{\mu}(f^*({\bm{w}})-f^*({\bm{w}}^\star))$$

The following corollary will be important to derive screening rules for penalized problems in Section \[sec:screening\_penalized\], as well as box-constrained problems (Section \[subsec:box\_screening\]).

\[cor:Dgaprestriction\] We consider the problem setup and , and assume $f$ is $\mu$-smooth. Then $${\left\lVert{\bm{w}}- {\bm{w}}^\star\right\rVert}^2 \le \frac{2}{\mu}{G}({\bm{x}}).$$ Here ${G}({\bm{x}})$ is the duality gap function as defined in equation .

The following two results hold for general constrained optimization problems of the form , where $g$ is the indicator function of a constraint set ${\mathcal{C}}\subset{\mathbb{R}}^n$ and hence are useful for deriving screening rules for such problems.

\[lem:FWgaprestriction\] Consider problem and assume that $f$ is 