Variational cost law: over proposals r, (Σ_j √(E[a_j] c_j))² ≤ (Σ_j r_j c_j)·E[Σ_j a_j/r_j], attained by r_j ∝ √(E[a_j]/c_j)

For all $\htmlData{mx=1}{I, J \in \htmlData{mx=0}{\operatorname{Type}}}$, $\htmlData{mx=3}{\mathit{lam} : \htmlData{mx=0}{I} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, $\htmlData{mx=5}{a : \htmlData{mx=0}{I} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{J} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ and $\htmlData{mx=3}{c, r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, if $\htmlData{mx=4}{\forall i,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\mathit{lam}}(\htmlData{mx=1}{i})}}$, $\htmlData{mx=5}{\forall i, j,\ 0 \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{\htmlData{mx=0}{a}(\htmlData{mx=1}{i}, \htmlData{mx=2}{j})}}$, $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{c}(\htmlData{mx=1}{j})}}$ and $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{I}}$, $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{J}}$), then