Theorem 7.6 — $Z(x)=\bar c x+\mathcal Q(x)$ is convex on $K$, and either finite on $K$ or identically $-\infty$

Consider a stochastic program with fixed recourse whose random element $\xi=(c,q,p,T)$, with law a probability measure $\mu$, satisfies the weak covariance condition (Definition 2.2). Let $Z(x)=\bar c x+\mathcal Q(x)$ be the objective of the deterministic equivalent program ((3.1), with the paper's integral), and $K=K_1\cap K_2$ its feasible region, where $K_1=\{x: Ax=b,\ x\ge0\}$ and $K_2$ is the induced constraint set of Corollary 4.5. Then one of the following holds (they exclude each other unless $K=\emptyset$):

1. $Z$ is finite on $K$ ($-\infty<Z(x)<+\infty$ for every $x\in K$) and convex on $K$:

which includes the convexity of $K$; 2. $Z(x)=-\infty$ for every $x\in K$.

This is the structural fact about the deterministic equivalent program: it is a convex program, and it cannot be finite at some feasible points and $-\infty$ at others.

Formalization Note "Convex" for an extended-real function is folded into the dichotomy: a function identically $-\infty$ is convex, and on the finite branch the statement is ConvexOn ℝ K (fun x => (Z x).toReal), where toReal is exact because finiteness is asserted in the same branch. The identity $E\{c(\xi)x+Q(x,\xi)\}=\bar cx+\mathcal Q(x)$ is the paper's definition (3.1) and is built into $Z$. The standing full-rank assumption on $W$ (p. 312) is not needed for this statement and is omitted, which makes it stronger.