Theorem 7.5 — iterating the Lovász–Schrijver operator $p$ times reaches the integer hull, $N^p(K) = \mathrm{conv}(K_0)$

This is Theorem 7.5 of Balas's Disjunctive Programming (cited to Lovász and Schrijver [99]). Let $K = \{x : \tilde A x \ge \tilde b\}$ be the LP relaxation of a mixed 0-1 program with $p = |N'|$ 0-1 variables, $K_0 = K \cap \{x_j \in \{0,1\},\ j \in N'\}$, and $N^1(K) = N(K)$, $N^t(K) = N(N^{t-1}(K))$. Then

The iterates are given by linear systems $(A_t, b_t)$, $t = 0,\dots,p$, with $\{x : A_0 x \ge b_0\} = K$ and $\{x : A_{t+1}x \ge b_{t+1}\} = N(\{x : A_t x \ge b_t\})$, every system containing the bound rows $x \ge 0$, $x_j \le 1$ ($j \in N'$).

Formalization Note. The retired version (Open) allowed systems without the bound rows, the gap that falsified Theorems 7.1, 7.4, 7.6 and 7.7 of the same chapter; with no rows, $N(K) = \mathbb R^n$ and the claim fails already for $p = 1$. 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. Each $N^t(K) \subseteq K$ lies in the box, so a system for it containing the bound rows always exists (append them as redundant rows); with them the linearized system defining $N$ coincides with Lovász and Schrijver's representation-free cone definition.