Proposition 2 — dom(6_ε) = dom(6) for small ε, and the objectives of (6_ε) converge uniformly to that of (6)

Let an instance of the two-stage robust binary program with constraint uncertainty be given, with $\mathcal X\subseteq\{0,1\}^N$, $\mathcal Y\subseteq\{0,1\}^M$ and $\Xi=\{\xi:A\xi\le b\}$ nonempty and bounded, and let $K\in\mathbb N$ be the number of policies. Denote by $\varphi$ and $\varphi_\varepsilon$ the objective functions of problems (6) and $(6_\varepsilon)$, and by $\mathrm{dom}(6)$ and $\mathrm{dom}(6_\varepsilon)$ their effective domains: the decisions $(x,\{y^k\}_{k\in\mathcal K})\in\mathcal X\times\mathcal Y^K$ at which the objective value is finite (not $+\infty$). Then:

1. $\mathrm{dom}(6_\varepsilon)=\mathrm{dom}(6)$ for all sufficiently small $\varepsilon>0$: there is $\varepsilon_0>0$ with

2. over the effective domain, $\varphi_\varepsilon$ converges uniformly to $\varphi$ as $\varepsilon\downarrow0$: for every $\kappa>0$ there is $\varepsilon_0>0$ such that for all $\varepsilon\in(0,\varepsilon_0]$ and all $(x,\{y^k\})\in\mathrm{dom}(6)$ the values $\varphi(x,\{y^k\})$ and $\varphi_\varepsilon(x,\{y^k\})$ are real numbers with

Since (6) is equivalent to the K-adaptability problem $\mathcal P_K$ (Proposition 1), this justifies solving the problems $(6_\varepsilon)$, whose uncertainty sets are closed, in place of $\mathcal P_K$.

Formalization Note Objective values are extended reals; "finite (i.e., do not evaluate to $+\infty$)" is read as $<+\infty$, as the paper says. Part (ii) is stated with a single threshold $\varepsilon_0$ that works for every decision of $\mathrm{dom}(6)$ at once, which is what "uniformly" means; since by (i) the two domains coincide for small $\varepsilon$, "over their effective domains" is the domain of (6). The remark after the proposition (convergence of the optimal values) is not part of the statement.