$\mu$-strongly convex over a bounded set ${\mathcal{C}}$. Then it holds that $$\label{eq:FWgaprestriction}
{\left\lVertA{\bm{x}}-A{\bm{x}}^\star\right\rVert}_2^2 \le \frac{1}{\mu}{G_{\mathcal{C}}}({\bm{x}}),$$ where ${\bm{x}}^\star$ is an optimal solution and ${G_{\mathcal{C}}}$ is the Wolfe-Gap function of $f$ over the bounded set ${\mathcal{C}}$.

\[cor:restric:smooth\] Assuming $f$ is $L$-smooth as well as $\mu$-strongly convex over a bounded set ${\mathcal{C}}$, we have $${\left\lVert\nabla f(A{\bm{x}})-\nabla f(A{\bm{x}}^\star)\right\rVert} \le \frac{L}{\sqrt{\mu}}\sqrt{{G_{\mathcal{C}}}({\bm{x}})}$$

Screening Rules for Constrained Problems {#sec:screening_constrained}
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In the following, we will develop screening rules for constrained optimization problems of the form , by exploiting the structure of the constraint set for a variety of sparsity-inducing problems. First of all, we give a general lemma which we will be using in rest of the paper to derive screening rules when any of the function in \[eq:primal\] and \[eq:dual\] is indicator function.

\[lem:general\_constrained\_lemma\] For general constrained optimization $\min_{{\bm{x}}\in {\mathcal{C}}} f(A{\bm{x}})$, the optimality condition  gives rise to the following optimality rule at the optimal point: $$\label{eq:general_constrained_rule}
(A{\bm{x}}^\star)^\top {\bm{w}}^\star = \min_{{\bm{z}}\in {\mathcal{C}}} (A{\bm{z}})^\top {\bm{w}}^\star \vspace{-1mm}$$

The above equation also suggest that ${\bm{x}}^\star = \operatorname*{arg\,min}_{{\bm{z}}\in {\mathcal{C}}}(A{\bm{z}})^\top {\bm{w}}^\star$. Lemma \[lem:general\_constrained\_lemma\] is very crucial in further deriving screening rules for constrained optimization problem as well as norm penalized problems whose conjugate is indicator function of the dual norm.

Simplex Constrained 