Theorem 2, p. 11 — with nonempty Type 2 ambiguity sets, the stage Bellman equation (2) equals the single minimization (11)

Consider one stage $t$ of the Bellman equations (2) under the Type 2 ambiguity set (10): a stage feasible set $S=X_t(x_{t-1},\xi_t)$ of pairs $(x_t,y_t)$ with binary state $x_t\in\{0,1\}^I$, a stage cost $g_t$, a finite support $\{\xi^k_{t+1}\}_{k=1}^K\subset\mathbb R^J$, next-stage values $Q^k_{t+1}(x_t)$, and a decision-dependent mean $\mu(x_t)$ and covariance $\Sigma(x_t)$. Suppose that for every feasible $x_t$ (every $x_t$ with some $y_t$ such that $(x_t,y_t)\in S$) the ambiguity set $\mathcal P^{D_2}_{t+1}(x_t)$ is nonempty. Then for every real $q$,

holds (with both the minimum and the inner maxima attained) if and only if $q$ is the attained minimum of

where $s\in\mathbb R$, $u\in\mathbb R^J$, $Y\in\mathbb R^{J\times J}$ and $A\bullet B=\operatorname{trace}(A^\top B)$.

In words: the min–max Bellman equation of stage $t$ can be rewritten as one minimization in which the worst-case expectation is replaced by its LP dual. Under the affine maps (12) and McCormick envelopes for the binary state this becomes the mixed-integer linear program that the paper's SDDiP algorithm solves.

Formalization Note The theorem holds for arbitrary $S$ and $g_t$, so the page's standing assumptions that $g_t$ is linear and $X_t$ is a nonempty compact mixed-integer polyhedron are not imposed (a disclosed generalization); the binary state is kept as the hypothesis hbin, which the statement does not need. The next-stage values are an arbitrary function of the state and the scenario. The ambiguity set includes $p\ge0$ ((C-18e) of the proof; (10) as printed omits it). The equality "$Q_t=$ (11)" is stated as: $q$ is the least stage value if and only if $q$ is the least value of (11), so both the existence of the minima and their equality are asserted.