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Program."],"published-print":{"date-parts":[[2022,7]]},"abstract":"Abstract<\/jats:title>In this paper we reconsider a known technique for constructing strong MIP formulations for disjunctive constraints of the form $$x \\in \\bigcup _{i=1}^m P_i$$<\/jats:tex-math>\n \n x<\/mml:mi>\n \u2208<\/mml:mo>\n \n \u22c3<\/mml:mo>\n \n i<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n m<\/mml:mi>\n <\/mml:msubsup>\n \n P<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where the $$P_i$$<\/jats:tex-math>\n \n P<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are polytopes. The formulation is based on the Cayley Embedding of the union of polytopes, namely, $$Q := \\mathrm {conv}(\\bigcup _{i=1}^m P_i\\times \\{\\epsilon ^i\\})$$<\/jats:tex-math>\n \n Q<\/mml:mi>\n :<\/mml:mo>\n =<\/mml:mo>\n conv<\/mml:mi>\n (<\/mml:mo>\n \n \u22c3<\/mml:mo>\n \n i<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n m<\/mml:mi>\n <\/mml:msubsup>\n \n P<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \u00d7<\/mml:mo>\n \n {<\/mml:mo>\n \n \u03f5<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msup>\n }<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where $$\\epsilon ^i$$<\/jats:tex-math>\n \n \u03f5<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the i<\/jats:italic>th unit vector in $${\\mathbb {R}}^m$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n m<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Our main contribution is a full characterization of the facets of Q<\/jats:italic>, provided it has a certain network representation. In the second half of the paper, we work-out a number of applications from the literature, e.g., special ordered sets of type 2, logical constraints, the cardinality indicating polytope, union of simplicies, etc., along with a more complex recent example. Furthermore, we describe a new formulation for piecewise linear functions defined on a grid triangulation of a rectangular region $$D \\subset {\\mathbb {R}}^d$$<\/jats:tex-math>\n \n D<\/mml:mi>\n \u2282<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> using a logarithmic number of auxilirary variables in the number of gridpoints in D<\/jats:italic> for any fixed d<\/jats:italic>. The series of applications demonstrates the richness of the class of disjunctive constraints for which our method can be applied.<\/jats:p>","DOI":"10.1007\/s10107-021-01652-z","type":"journal-article","created":{"date-parts":[[2021,4,16]],"date-time":"2021-04-16T03:33:40Z","timestamp":1618544020000},"page":"831-869","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Ideal, non-extended formulations for disjunctive constraints admitting a network representation"],"prefix":"10.1007","volume":"194","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2759-1264","authenticated-orcid":false,"given":"Tam\u00e1s","family":"Kis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mark\u00f3","family":"Horv\u00e1th","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,4,16]]},"reference":[{"key":"1652_CR1","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s10107-020-01474-5","volume":"193","author":"R Anderson","year":"2020","unstructured":"Anderson, R., Huchette, J., Ma, W., Tjandraatmadja, C., Vielma, J.P.: Strong mixed-integer programming formulations for trained neural networks. 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