Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 339 / 1 / c / Solution

Write $z_i(x)=⟨a_i,x⟩+b_i$. The log-sum-exp function is convex and composition with the affine functions $z_i$ preserves convexity, so $f_{\beta }$ is convex. Directly, its Hessian matrix will also be shown positive semidefinite in part d.

Let $M=f(x)={\max }_i{z}_i(x)$. Factoring $e^{\beta M}$ out of the sum givesAt least one term in the sum is $1$, while every term is at most $1$. Thereforeand henceThus $f_{\beta }$ is a uniform smooth maximum of the affine pieces of $f$.