{"ID":5935672,"CreatedAt":"2026-07-07T01:22:02.77346169Z","UpdatedAt":"2026-07-07T02:10:06.972658124Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.03460","arxiv_id":"2607.03460","title":"On disjunction convex hulls for generalized cross polytopes","abstract":"We continue the study of the natural polytope $\\mathcal{D}$ in $\\mathbb{R}^{n+d}$ associated with the disjunction of a set of $n+1$ polytopes in $\\mathbb{R}^d$, managed by $n$ binary variables. Already $\\mathcal{D}$ had been characterized for arbitrary $n\\geq 1$ and (i) $d\\in\\{1,2\\}$, and (ii) for a broad generalization of hyper-rectangles. In both cases, the complete characterization employs full optimal big-M lifting. Here, we give a complete description of $\\mathcal{D}$ for the case of $n=1$ and arbitrary $d$, when the (two) polytopes are arbitrary generalized cross polytopes. Furthermore, we characterize when our complete description employs only optimal big-M lifting. For $n\u003e1$, we generalize the family of facet-describing inequalities used for $n=1$. Finally, we carry out some computational experiments demonstrating the value of our theoretical results.","short_abstract":"We continue the study of the natural polytope $\\mathcal{D}$ in $\\mathbb{R}^{n+d}$ associated with the disjunction of a set of $n+1$ polytopes in $\\mathbb{R}^d$, managed by $n$ binary variables. Already $\\mathcal{D}$ had been characterized for arbitrary $n\\geq 1$ and (i) $d\\in\\{1,2\\}$, and (ii) for a broad generalizatio...","url_abs":"https://arxiv.org/abs/2607.03460","url_pdf":"https://arxiv.org/pdf/2607.03460v1","authors":"[\"Yushan Qu\",\"Jon Lee\"]","published":"2026-07-03T16:17:06Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}