Theorem 4.6 — $\bar z\in N$ iff $\bar z$ minimizes the lexicographic weighted Tchebycheff program for some $\lambda\in\bar\Lambda$

Let $Z\subseteq\mathbb R^k$ ($k\ge1$) be a compact set of feasible criterion vectors with nondominated set $N$, and let $z^*$ be an ideal criterion vector for $Z$. For every $\bar z\in Z$,

where $\bar\Lambda=\{\lambda\in\mathbb R^k\mid\lambda_i\ge0,\ \sum_i\lambda_i=1\}$ and the lexicographic program first minimizes $\alpha=\max_i\lambda_i(z^*_i-z_i)$ and then, among the first-stage minimizers, minimizes $e^{\mathsf T}(z^*-z)=\sum_i(z^*_i-z_i)$.

The lexicographic weighted Tchebycheff program therefore characterizes the nondominated set for an arbitrary (in particular nonconvex, continuous) feasible region, without choosing an augmentation parameter $\rho$.

Formalization Note $S$, the objectives $f_i$ and the variable $\alpha$ are eliminated: $Z$ is the set of criterion vectors. Compactness of $Z$ is an added hypothesis, used by the direction $\Rightarrow$ (through Theorem 4.5); the paper assumes $S$ bounded. It is not removable: for a bounded but non-closed $Z$ the direction $\Rightarrow$ can fail (example in the Formalization Note of Theorem 4.5). The direction $\Leftarrow$ holds without it. The printed priority "$P_1<<<P_2$" is reversed relative to the paper's two-stage description on p. 336, which is what is encoded.