Theorem 2.18 — facets of $\mathrm{cl\,conv}(F)$ are the extreme rays of $W_0$

This is Theorem 2.18 of Balas's Disjunctive Programming, the culminating result of Chapter 2: the facets of the closed convex hull of a union of polyhedra are exactly the extreme rays of the explicit cone $W_0$.

Let $P_h := \{x \in \mathbb{R}^n : A_h x \ge b_h\}$ ($h \in Q$), $F := \bigcup_{h\in Q} P_h$, $Q^* := \{h : P_h \ne \emptyset\}$, and

Assume $F$ is full-dimensional, $\dim F = n$. Then for $\alpha \ne 0$ and $\alpha_0 \ne 0$:

where "$\alpha x \ge \alpha_0$ defines a facet" means that the inequality is valid for $\mathrm{cl\,conv}(F)$ and $\mathrm{cl\,conv}(F) \cap \{x : \alpha x = \alpha_0\}$ is a facet of $\mathrm{cl\,conv}(F)$.

Formalization Note. The retired version expressed "defines a facet" by IsFacet applied to $\mathrm{cl\,conv}(F) \cap \{\alpha x = \alpha_0\}$, which does not say on which side of the hyperplane $\mathrm{cl\,conv}(F)$ lies; it was refuted by $F = [0,1]$, $\alpha = 1$, $\alpha_0 = 1$ ($x \le 1$, not $x \ge 1$, is the valid facet inequality). The new predicate DefinesFacetGE (module Disjunctive_Polarity_FacetDefining) adds validity of $\alpha x \ge \alpha_0$. The hypothesis $\alpha \neq 0$ is made explicit: an inequality $\alpha x \ge \alpha_0$ in the book's sense has a nonzero normal, and without it the statement fails, since for $F = [0,\infty) \subseteq \mathbb{R}$ the cone $W_0 = \{\alpha \ge 0,\ \alpha_0 \le 0\}$ has the extreme ray $(0,-1)$, while $0 \cdot x \ge -1$ defines no facet. With $\alpha \ne 0$ the statement is the standard fact that $W_0$ is the cone of valid inequalities of the full-dimensional polyhedron $\mathrm{cl\,conv}(F)$ (Theorem 1.2 / Corollary 2.17), whose extreme rays are its facet inequalities and possibly $(0,-1)$. Full-dimensionality is PolyDim F = n (affine hull of $F$ is $\mathbb{R}^n$).