Proof of Theorem 4.5 — $\bar z\in\Phi(\hat\alpha)$, and no other nondominated $z$ lies in $\Phi(\hat\alpha)$

Let $Z\subseteq\mathbb R^k$ ($k\ge1$) be a set of criterion vectors with nondominated set $N$, let $z^*$ be an ideal criterion vector for $Z$, and let $\bar z\in N$. Let $\bar\lambda$ be the weights of eq. (4.3) for $\bar z$, and let $\hat\alpha$ be the minimal value of the weighted Tchebycheff program with weights $\bar\lambda$, i.e. the least value of $\max_i\bar\lambda_i(z^*_i-z_i)$ over $z\in Z$. With

we have

This is the key step in the proof of Theorem 4.5: among nondominated vectors, $\bar z$ is the only one reached by the first stage of the lexicographic program with weights $\bar\lambda$. Dominated vectors may also lie in $\Phi(\hat\alpha)$; removing them is the task of the second stage.

Formalization Note $Z$ is an arbitrary set (no finiteness or compactness is assumed). The existence of the minimum $\hat\alpha$ is taken as the hypothesis that $\hat\alpha$ is the least element of the image of $Z$ under the first-stage objective. The claim is stated for $z\in N$ only, as on the page; the paper's next sentence ("the associated weighted Tchebycheff program has a unique solution") is not formalized, because a dominated vector $z$ with $z_j=z^*_j=\bar z_j$ can tie with $\bar z$ in the first stage (e.g. $Z=\{(5,3),(5,1),(1,10)\}$, $z^*=(5,10)$, $\bar z=(5,3)$, $\bar\lambda=(1,0)$).