{"paper":{"title":"On finite generation and infinite convergence of generalized closures from the theory of cutting planes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.OC","authors_text":"Gennadiy Averkov","submitted_at":"2011-06-08T09:22:17Z","abstract_excerpt":"For convex sets $K$ and $L$ in ${\\mathbb{R}}^d$ we define $R_L(K)$ to be the convex hull of all points belonging to $K$ but not to the interior of $L$. Cutting-plane methods from integer and mixed-integer optimization can be expressed in geometric terms using functionals $R_L$ with appropriately chosen sets $L$. We describe the geometric properties of $R_L(K)$ and characterize those $L$ for which $R_L$ maps polyhedra to polyhedra. For certain natural classes ${\\mathcal{L}}$ of convex sets in ${\\mathbb{R}}^d$ we consider the functional $R_{\\mathcal{L}}$ given by $R_{\\mathcal{L}}(K):= \\bigcap_{L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.1526","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}