Proposition 7.5 — $Q(x,\xi)$ is convex polyhedral in $x$ on $K_2$, concave polyhedral in $q$, convex polyhedral in $(p,T)$

Consider the recourse function $Q(x,\xi)=\min\{q(\xi)y\mid Wy=p(\xi)-T(\xi)x,\ y\ge0\}$ of a stochastic program with fixed recourse $W$, the support $\tilde\Xi$ of the law $\mu$ of $\xi$, and the induced constraint set $K_2$ (Corollary 4.5). Then:

1. In $x$. For each $\xi\in\tilde\Xi$, either $x\mapsto Q(x,\xi)$ is a finite convex polyhedral function on $K_2$ (a maximum of finitely many affine functions of $x$ there), or $Q(x,\xi)=-\infty$ for every $x\in K_2$.
2. In $q$. Fix $x$, $p$ and $T$ with $p-Tx\in\operatorname{pos}W$. Then $q\mapsto Q$ is a finite concave polyhedral function (a minimum of finitely many affine functions of $q$) on

the set of cost vectors for which the dual of the recourse problem is feasible (Corollary 7.4). 3. In $(p,T)$. Fix $q$ and $x$. On the set $\{(p,T) \mid p-Tx\in\operatorname{pos}W\}$, either $(p,T)\mapsto Q$ is a finite convex polyhedral function (a maximum of finitely many affine functions of $(p,T)$), or it is identically $-\infty$.

Part 1 is what Theorem 7.6 uses to obtain the convexity of the expected recourse $\mathcal Q$ and its finite-or-$-\infty$ dichotomy; parts 2 and 3 describe the dependence on the random data.

Formalization Note In parts 2 and 3, $Q$ is the platform's KallMayer.Recourse.PointwiseRecourse W T p q x evaluated at arbitrary data $(q,p,T)$. Parts 1 and 3 are stated as the dichotomy of Corollary 7.3, which is what "polyhedral" means for a function that may take the value $-\infty$; part 2 is stated as Corollary 7.4, the corollary the paper's proof invokes for it (the primal problem is feasible by hypothesis, so the $+\infty$ branch of Corollary 7.4 does not occur). The paper remarks (p. 329) that the proposition extends to all $x\in\mathbb R^n$ under the $\pm\infty$ conventions; the printed statement, on $K_2$ and for $\xi\in\tilde\Xi$, is what is formalized.