Theorem 2.2, p. 8 — for continuous concave or quadratic L, worst-case CVaR ≤ 0 iff inf_P P(L ≤ 0) ≥ 1 − ε

Let $\mathcal P$ be the set of all probability distributions on $\mathbb R^k$ with mean vector $\mu$ and covariance matrix $\Sigma\succ0$, and let $\epsilon\in(0,1)$. Let $L:\mathbb R^k\to\mathbb R$ be a continuous loss function that is either

1. concave in $\xi$, or
2. (possibly nonconcave) quadratic in $\xi$.

Then the following equivalence holds:

Here $\mathbb P\text{-}\mathrm{CVaR}_\epsilon(L(\tilde\xi))=\inf_{\beta}\{\beta+\frac1\epsilon\mathbb E_{\mathbb P}((L(\tilde\xi)-\beta)^+)\}$. The implication from left to right holds for every loss; the theorem says that for these two classes of losses the worst-case CVaR constraint, which is a tractable conservative approximation of the distributionally robust chance constraint, is in fact exact.

Formalization Note The worst-case CVaR is extended-real valued, so a loss with infinite CVaR under some $\mathbb P\in\mathcal P$ is not mapped to a junk value. $\mathcal P$ consists of all probability measures with finite second moments and the given mean and covariance. A quadratic loss is $\xi^\top Q\xi+q^\top\xi+q^0$ with no definiteness or symmetry imposed on $Q$; concavity is on all of $\mathbb R^k$. The standing assumptions $\Sigma\succ0$ and $\epsilon\in(0,1)$ of §2 are hypotheses.