Theorem 7.7 — if $Z$ is bounded on $K$, then $Z$ is Lipschitz on $K$

Consider a stochastic program with fixed recourse whose random element $\xi$, with law a probability measure $\mu$, satisfies the weak covariance condition (Definition 2.2), and whose recourse matrix $W$ has full row rank (the paper's standing assumption, p. 312). Let $Z(x)=\bar cx+\mathcal Q(x)$ and $K=K_1\cap K_2$ be as in the model. Suppose $Z$ is bounded on $K$, i.e. $Z(x)>-\infty$ for every $x\in K$. Then $Z$ is finite on $K$ and satisfies a Lipschitz condition there: there is a constant $\bar B$ such that

where $\|\cdot\|$ is the Euclidean norm of $\mathbb R^n$.

Together with Theorem 7.6 this shows that, whenever the deterministic equivalent program is not identically $-\infty$, its objective is a finite, convex, Lipschitz function on the feasible region; this is the regularity used for the stability result, Theorem 8.11.

Formalization Note The paper writes "$Z(x)$ is bounded on $K$". By Theorem 7.6, $Z$ is either finite or identically $-\infty$ on $K$, and the proof opens with "if $\mathcal Q(x)>-\infty$ on $K_2$"; the hypothesis is therefore read as $Z>-\infty$ on $K$, which is weaker than a two-sided bound $|Z|\le M$ and makes the theorem stronger. Finiteness on $K$ is part of the conclusion, so that toReal is exact. The constant $\bar B$ may depend on the instance. The Euclidean norm is written as $\sqrt{\sum_i (x_i-x^0_i)^2}$.