Lemma 8.9 — a finite program with convex objective Lipschitz on a polyhedron is stable

Let $A$ be a real $m\times n$ matrix, $b\in\mathbb R^m$, $P\subseteq\mathbb R^n$ a convex polyhedron (finitely many linear inequalities), and $f:\mathbb R^n\to\mathbb R$ a function that is convex on $P$ and Lipschitz on $P$. Consider the mathematical program (8.10),

where $f$ is regarded as $+\infty$ outside $P$, so that the feasible set is $P\cap\{x: Ax=b,\ x\ge0\}$. If (8.10) is finite, i.e. its optimal value $v=\inf\{f(x): x\in P,\ Ax=b,\ x\ge0\}$ is a real number, then (8.10) is stable: there is $\pi\in\mathbb R^m$ with

This is the purely convex-analytic step behind Theorem 8.11 (the paper cites it to Walkup and Wets, 1969): a Lipschitz convex objective on a polyhedral domain rules out the infinite-slope behaviour of the perturbation function that destroys stability.

Formalization Note "Convex and Lipschitz on a polyhedron" is read as: the objective's domain is the polyhedron $P$ (the objective is $+\infty$ off $P$), and on $P$ it is finite, convex and Lipschitz. This is how the lemma is applied to $Z$, whose domain is $K_2$; a function finite and convex on all of $\mathbb R^n$ would make the Lipschitz hypothesis superfluous. Lipschitz is LipschitzOnWith for the sup norm, equivalent to the Euclidean one for the existence of a constant. "Stable" is the Kuhn–Tucker form of Definition 8.1(iv) given in the definition file: the dual obtained by perturbing $b$ is solvable and has no duality gap.