Theorem 7.6 — the Sherali–Adams hierarchy reaches the integer hull, $K_p = \mathrm{conv}(K_0)$

This is Theorem 7.6 of Balas's Disjunctive Programming (cited to Sherali and Adams [112]). Let $K = \{x \in \mathbb R^n : \tilde A x \ge \tilde b\}$ be the LP relaxation of a mixed 0-1 program whose 0-1 variables are indexed by $N'$, $|N'| = p$, where the system $\tilde A x \ge \tilde b$ contains the bound rows $x \ge 0$ and $x_j \le 1$ ($j \in N'$), and let $K_0 = K \cap \{x : x_j \in \{0,1\},\ j \in N'\}$. Let $K_t$ be the level-$t$ Sherali–Adams relaxation: multiply every row of $\tilde A x \ge \tilde b$ by every product $\prod_{j \in J_1} x_j \prod_{j \in J_2}(1 - x_j)$ with $J_1, J_2 \subseteq N'$ disjoint, $|J_1 \cup J_2| = t$, linearize the monomials with moment variables (using $x_j^2 = x_j$ for $j \in N'$) and project onto $x$. Then

Formalization Note. The retired version allowed an arbitrary system $Ax \ge b$; with no rows, $K_p = \mathbb R^n$ while $\mathrm{conv}(K_0)$ lies in the unit box. Throughout Chapter 7 the book works with $K = \{x \in \mathbb R^n : Ax \ge b,\ x \ge 0,\ x_j \le 1,\ j \in N'\} = \{x : \tilde A x \ge \tilde b\}$: the bound constraints are rows of the system that every construction multiplies. This is now the explicit hypothesis HasBoundRows A b N' (the system contains the rows $x_k \ge 0$ for every $k$ and $-x_j \ge -1$ for every $j \in N'$); it is satisfiable with nonempty $K$ (e.g. the unit box) and excludes the disproof's instance with no rows. The definitions KtSet/IsXt/RowNLt are unchanged: they multiply all rows of the given system, hence the bound rows too, exactly as the book multiplies $\tilde A x \ge \tilde b$.