Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 7H (Optimisation)

For $0\le t\le 1$ and $x,y\in {\mathbb{R}}^d$, convexity givesTaking the maximum over $i$ and then bounding each term by the corresponding endpoint maxima yieldsThus the finite pointwise maximum of convex functions is convex. Summing the original inequalities over $i$ proves that ${\sum }_i{f}_i$ is convex as well.

Because $x\mapsto c^{T}x$ is a linear map,Hence composition of a convex function with an affine map is convex, and $g$ is convex.

The softplus functionis convex becausePart (b) therefore shows that every $\beta \mapsto h(a_i^T\beta )$ is convex. The absolute value function is convex, so $\beta \mapsto |{\beta }_j|$ is convex for every coordinate $j$. Finally, part (a) says that a finite sum of convex functions is convex. Thereforeis convex. The second sum is the L1 norm regularizer.