0 
\ \Rightarrow \  x_i^\star =1 \ . $$

Box constrained opptimization problems arise very often in machine learning probelm. Hinge loss SVM happens to one of many special cases of box-constrained optimizaion problem.

#### Hinge Loss SVM.

The dual of the classical support vector machine with hinge loss, when not using a bias value, is a box-constrained problem. As a direct consequence of Theorem \[thm:box\_constrained\] we therefore obtain screening rules for SVM with hinge loss and no bias. The primal formulation of the SVM in this setting, for a regularization parameter $C>0$, is$$\begin{aligned}
 &\underset{{\bm{w}}\in{\mathbb{R}}^d,{\bm{\epsilon}}\in{\mathbb{R}}^n}{\min}  \tfrac{1}{2} {\bm{w}}^\top{\bm{w}}+ C {\mathbf{1}}^\top{\bm{\epsilon}}\\
 &\text{s.t.}  \quad  {\bm{w}}^\top{\bm{a}}_i \ge 1-\varepsilon_i  \ \ \forall i \in [n]  \\
\qquad  & \varepsilon_i \ge 0  \quad \forall i \in [n] \label{eq:primal:svm_hinge}
\end{aligned}$$

\[cor:SVMhinge\] For SVM with hinge loss and no bias as given in , we have the screening rules $$\begin{aligned}
&{\bm{a}}_i^\top A{\bm{x}}-{\left\lVert{\bm{a}}_i\right\rVert}_2\sqrt{2{G}({\bm{x}})} >0 \ \Rightarrow \ x_i^\star = 0 \ , \ \text{ and}\\
&{\bm{a}}_i^\top A{\bm{x}}+{\left\lVert{\bm{a}}_i\right\rVert}_2\sqrt{2{G}({\bm{x}})} <0 \ \Rightarrow \ x_i^\star = C \ .\end{aligned}$$ where ${\bm{x}}\in{\mathbb{R}}^n$ is any feasible dual point.

We get similar screening rules for hinge loss SVM as in [@Zimmert:2015ue] as well as in [@Shibagaki:2016ts]. The closest known result to our Corollary \[cor:SVMhinge\] for 