 Solution

For fixed $x$, put $z=r-r_0$. The polyhedral uncertainty set is equivalent to $-1\le Pz\le 1$, so the inner problem is the linear programThe variable $z$ is unrestricted in sign. The point $z=0$ is strictly feasible. If the optimum is finite, it is attained, so the infimum is a minimum. The use of an infimum also covers possible unbounded cases permitted by the printed assumptions; strict feasibility alone does not guarantee a finite objective.