Corollary 4 (corrected) — worst-case VaR for a diagonal covariance bound

Let $\mathcal P$ be the covariance ambiguity set (16) with box $[\underline{\boldsymbol q},\overline{\boldsymbol q}]$, $\underline{\boldsymbol q}\ge\mathbf 0$, mean $\boldsymbol\mu$ in the interior of the box, and diagonal covariance bound $\Sigma=\operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2)$ with $\sigma_i>0$. Let $\epsilon\in(0,1)$, $r=\frac{1-\epsilon}{\epsilon}$, $\boldsymbol q^u=\min\{\frac{1-\epsilon}{\epsilon}(\boldsymbol\mu-\underline{\boldsymbol q}),\overline{\boldsymbol q}-\boldsymbol\mu\}$ componentwise, and for $\theta\ge0$ let $S(\theta)=\{i\in S:\sigma_i^2>\theta\,q^u_i\}$ and $s(\theta)=r-\sum_{i\in S(\theta)}(q^u_i/\sigma_i)^2$. Then for every customer subset $S$

where $\Theta$ is the set of $\theta\ge0$ such that

1. $s(\theta)\ge0$ (the paper's restriction of the feasible region), and
2. $\sigma_i^2\sqrt{s(\theta)}\le q^u_i\sqrt{\sum_{k\in S\setminus S(\theta)}\sigma_k^2}$ for every $i\in S\setminus S(\theta)$.

Condition 2 says that the solution of (17) induced by $\theta$ — $q_i=q^u_i$ on $S(\theta)$, $q_i=\sigma_i^2\sqrt{s(\theta)}/\sqrt{\sum_{S\setminus S(\theta)}\sigma_k^2}$ on $S\setminus S(\theta)$, $q_i=0$ off $S$ — respects the upper bound $\boldsymbol q^u$. The result turns the quadratically constrained program of Theorem 7 into a one-parameter search, which the paper uses to evaluate the worst-case value-at-risk in time linear in $|S|$ once the ratios $q^u_i/\sigma_i^2$ are sorted.

This is a corrected statement. Corollary 4 as printed maximizes over all $\theta\ge0$ satisfying condition 1 only. That claim is false: for $n=1$, $S=\{1\}$, every large $\theta$ gives $S(\theta)=\emptyset$ and the value $\mu_1+\sigma_1\sqrt r$, whereas the worst-case value-at-risk is $\mu_1+\min\{q^u_1,\sigma_1\sqrt r\}$ (Theorem 7), which is smaller whenever $q^u_1<\sigma_1\sqrt r$; e.g. $\mu_1=1$, $\underline q_1=0$, $\overline q_1=1.2$, $\sigma_1=1$, $\epsilon=\frac12$ gives $2$ on the printed right-hand side, larger than $\overline q_1$, which bounds every value-at-risk. Condition 2 restores the equality: every $\theta\in\Theta$ induces a feasible point of (17), and the optimal solution of (17) is induced by some $\theta\in\Theta$.

Formalization Note As in Theorem 7, the paper writes "$\mathbb P$-VaR" without the level; the level $1-\epsilon$ used everywhere else in the paper is read in. Condition 2 is written without division so that it is meaningful when $S\setminus S(\theta)=\emptyset$ (then it is vacuous). The worst-case VaR and the program use the definitions covarianceSet, worstCaseVaR, qUpper, capSet, capSlack, diagObjective.