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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.

For every decision with finite worst-case value, the previous dual represents that value as a maximum over $\alpha ,\beta$. Maximizing it jointly with $x$ yields the robust linear optimization over the probability simplex formulation:Indeed every feasible triple gives a dual lower bound on that decision's worst-case revenue, while dual attainment supplies a triple reaching it. This argument only combines two maximizations after dualization; it does not interchange the original max and min.

If the probability simplex intersects $\mathrm{range}{P}^T$, this linear program is feasible and bounded above by ${\max }_i(r_0{)}_i$, and hence attains an optimum. Its maximizing $x$ is a robust optimizer. If the intersection is empty, every decision has worst-case value $-\infty$ and the linear program is infeasible, consistently with the extended-value convention. Full column rank of $P$ ensures every simplex decision has a finite inner optimum, but is not needed for the qualified equivalence.

Worst-case linear revenue over a polyhedral uncertainty set is optimized by the linear programHere ${\Delta }_n$ is the probability simplex. Finite decisions are exactly its intersection with $\mathrm{range}{P}^T$. If this intersection is empty, all decisions have worst-case value $-\infty$ and the displayed linear program is infeasible; its supremum over the empty feasible set is also $-\infty$. Full column rank of $P$ suffices to make all simplex decisions finite.