\ x_i^\star=0 \ .$$

Our general screening rules for simplex constrained problems as in Theorem \[thm:simplex\_smooth\] allows many practical implications. For example, new screening rules for squared loss SVM and minimum enclosing ball problem come as a direct consequence.

#### Squared Hinge Loss SVM.

The squared hinge-loss SVM problem in its dual form is formulated as $$\begin{aligned}
 \label{eq:sq:loss:svm}
\min_{{\bm{x}}\in {\triangle}} \ \big[\ f(A{\bm{x}}) := \tfrac12 {\bm{x}}^\top A^\top A{\bm{x}}\ \big] \vspace{-1mm}
 $$ over a unit simplex constraint ${\bm{x}}\in{\triangle}\subset{\mathbb{R}}^n$. Here for given data examples $\bar{\bm{a}}_1,\dots,\bar{\bm{a}}_n\in{\mathbb{R}}^d$ and corresponding labels $y_i\in\pm1$, the matrix $A$ collects the columns ${\bm{a}}_i=y_i\bar{\bm{a}}_i$, see e.g. [@Tsang:2005up]. We obtain the following novel screening rule for square loss SVM:

\[cor:square\_svm\] For the squared hinge loss SVM we have the screening rule $$\begin{aligned}
({\bm{a}}_i - A{\bm{x}})^\top A{\bm{x}}&>\sqrt{\max_{i}\; (A{\bm{x}}-{\bm{a}}_i)^\top A{\bm{x}}} \,\|{\bm{a}}_i - A{\bm{x}}\| \notag \\
& \Rightarrow \ x_i^\star = 0.\end{aligned}$$

#### Minimum Enclosing Ball.

The primal and dual for the minimum enclosing ball problem is given as the following pair of optimization formulations and respectively. $$\begin{aligned}
& \min_{{\bm{c}}\in{\mathbb{R}}^d,r\in{\mathbb{R}}} \ \ \ r^2  \quad \text{s.t.} \ \ {\left\lVert{\bm{c}}-{\bm{a}}_i\right\rVert}^2_2\le r^2 \quad \forall i \in [n] \label{eq:min_enclosing_ball} \\
\qquad &~ \min_{{\bm{x}}\in{\triangle}\subset{\mathbb{R}}^n} \ \ \ {\bm{x}}^\top A^\top A{\bm{x}}+ {\bm{c}}^\top {\bm{x}}\ , \label{eq:dualball}\end{aligned}$$ where ${\bm{c}}$ is a 