§4, p. 336 — the lexicographic program has a solution, and every solution is nondominated

Let $Z\subseteq\mathbb R^k$ ($k\ge1$), $z^*\in\mathbb R^k$ and weights $\lambda\in\bar\Lambda=\{\lambda\in\mathbb R^k\mid \lambda_i\ge0,\ \sum_i\lambda_i=1\}$. Then:

1. if $Z$ is compact and nonempty, the lexicographic weighted Tchebycheff program with weights $\lambda$ has a minimizer;
2. every minimizer $z$ of the lexicographic weighted Tchebycheff program with weights $\lambda$ is nondominated:

This makes precise the remark that if the first-stage minimization of $\alpha$ does not yield a nondominated vector, the second-stage minimization of $e^{\mathsf T}(z^*-z)$ moves to one that is.

Formalization Note Part 1 uses compactness of $Z$, an added hypothesis (the paper assumes $S$ bounded and speaks of minimizers). Part 2 needs neither compactness nor the ideal-vector property, and $z^*$ is arbitrary in both parts.