Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 339 / 1 / a

A feasible point lies in the probability simplex, so $r^{T}x={\sum }_i{x}_i{r}_i$ is a convex combination of the coordinates of $r$. Hence $r^{T}x\le r_{\max }:={\max }_i{r}_i$. Taking a coordinate vector $e_j$ with $r_j=r_{\max }$ attains equality. ThereforeMore generally, all optimal solutions are exactly the probability vectors supported on the maximizing indices: equality requires $x_i(r_{\max }-r_i)=0$ for every $i$. Thus ties give an entire optimal face, while a unique maximizing index gives a unique optimizer.