Theorem 3.6, p. 20 — X°^JCC ⊆ ⋃_{α>0} Z^JCC(α) ⊆ X^JCC

Let $\tilde\xi\in\mathbb R^k$ be a random vector whose distribution is known only through its mean $\mu$ and covariance $\Sigma\succ0$, and let $\mathcal P$ be the set of all probability distributions with these moments. Let $\epsilon\in(0,1)$ and consider $m\ge1$ constraints $y_i^0(x)+y_i(x)^\top\tilde\xi\le0$ whose coefficients depend affinely on a decision $x\in\mathbb R^n$. Define

1. the robust joint chance constraint set $\mathcal X^{\mathrm{JCC}}=\{x:\inf_{\mathbb P\in\mathcal P}\mathbb P(y_i^0(x)+y_i(x)^\top\tilde\xi\le0\ \forall i)\ge1-\epsilon\}$;
2. its strict version $\mathcal X^{\mathrm{JCC}}_\circ=\{x:\inf_{\mathbb P\in\mathcal P}\mathbb P(\bigcap_i\{y_i^0(x)+y_i(x)^\top\tilde\xi<0\})\ge1-\epsilon\}$;
3. the worst-case CVaR approximation with optimized scaling parameters $\mathcal Z^{\mathrm{JCC}}=\bigcup_{\alpha>0}\mathcal Z^{\mathrm{JCC}}(\alpha)$, where $\mathcal Z^{\mathrm{JCC}}(\alpha)=\{x:\sup_{\mathbb P\in\mathcal P}\mathbb P\text{-}\mathrm{CVaR}_\epsilon(\max_i\alpha_i(y_i^0(x)+y_i(x)^\top\tilde\xi))\le0\}$.

Then

The worst-case CVaR approximation of a distributionally robust joint chance constraint is thus essentially exact when the scaling parameters are treated as decision variables: it is squeezed between the robust constraint and its strict version.

Formalization Note The union is over componentwise strictly positive $\alpha$ (the page prints $\bigcup_{\alpha\in\mathcal S}$; $\mathcal S$ is read as $\mathcal A$ of p. 15). CVaR and the worst-case CVaR are extended-real valued, so no junk value of a real supremum enters. $m\ge1$ (implicit in the maximum over $i$), $\Sigma\succ0$ and $\epsilon\in(0,1)$ are hypotheses. No nondegeneracy assumption on the constraint functions is made.