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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.