Theorem 3, p. 13 — (6) ⟹ (7) ⟹ (3) for the naïve approximation, the infimal convolution bound and the distributionally robust constraint; equivalence if J = 1

Let $\mathcal P$ be the standardized ambiguity set (4) with confidence sets $\mathcal C_i = \{(z,u) : C_iz + D_iu \preccurlyeq_{\mathcal K_i} c_i\}$, $i \in \mathcal I$, and assume:

1. every $\mathcal K_i$ is a proper cone and $0 \le \underline p_i \le \overline p_i \le 1$;
2. (C1): $\mathcal C_I$ is bounded and $\underline p_I = \overline p_I = 1$;
3. (C2): some $\mathbb P \in \mathcal P$ has $\mathbb P[(\tilde z,\tilde u)\in\mathcal C_i] \in (\underline p_i,\overline p_i)$ whenever $\underline p_i < \overline p_i$;
4. (C3): $v(x,z) = \max_{l}\,(S_lz + s_l)^\top x + \mathbf t_l^\top z + t_l$;
5. $\{\mathcal I_j\}_{j\in\mathcal J}$ is a partition of $\mathcal I$ (all blocks nonempty) satisfying (N′), with outer approximations $\mathcal P^j$;
6. (added) the uniform first-moment bound: for each $j$ there is $M_j$ with $\mathbb E_{\mathbb P}\|\tilde z\| \le M_j$ for all $\mathbb P\in\mathcal P^j$.

Then for every $x \in \mathbb R^N$ and $w \in \mathbb R$, the naïve approximation (6), the infimal convolution bound (7) and the distributionally robust constraint (3),

satisfy

and if $J = 1$, then (3) implies both (7) and (6), so all three are equivalent.

Both (6) and (7) are thus conservative surrogates for (3) when the nesting condition (N) of Theorem 1 fails, and (7) is never looser than (6).

Formalization Note The uniform first-moment bound is not in the paper. The printed proof of (6) $\Rightarrow$ (7) uses that $\sup_{\mathbb P\in\mathcal P^j}\mathbb E_{\mathbb P}[v(0,\tilde z)]$ is finite, which fails in general: with $P = 1$, $\mathcal C_1 = [0,\tfrac12]$ (probability $\ge \tfrac12$), $\mathcal C_2 = [0,1]$ (probability 1), the singleton partition, $v(x,z) = z$ and $w = 1$, (6) holds while (7) fails because $\mathcal P^1$ contains $\tfrac12\delta_0 + \tfrac12\delta_M$ for every $M$. The bound holds, for example, when every block contains a bounded confidence set with $\underline p_i = 1$. Indices are 0-based: $\mathcal I$ is Fin (nI + 1), $\mathcal C_I$ is Fin.last nI, $\mathcal J$ is Fin nJ, and the partition is a surjective block map. Distributions in $\mathcal P$ and $\mathcal P^j$ have finite first moments. (6) and (7) are EReal-valued; (3) is in its $\forall$-form. The hypotheses (N′), (C1), proper cones and $0\le\underline p_i\le\overline p_i\le1$ are the paper's setting and are not used by the proof.