I have a question and don't know whether there is a (good) solution.
Given two distinct integer vectors $ p \neq q \in \mathbb{Z}_{+}^n $ with $ p_i \le q_i $ for each component. Let $ K \in \mathbb{Z}_{+} $ be given. Is there a good way, to enforce $ q_{\beta}-p_{\alpha} \le K $ by means of (a few number of) linear inequalities, where the indices are defined as: $ \beta=\sup\left\lbrace 1\le i \le n \, : \, q_i \neq p_i \right\rbrace $ and $ \alpha=\inf\left\lbrace 1\le i \le n \, : \, q_i \neq p_i \right\rbrace $.