{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T08:54:39Z","timestamp":1772614479507,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2011,10,28]],"date-time":"2011-10-28T00:00:00Z","timestamp":1319760000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2013,2]]},"DOI":"10.1007\/s10107-011-0496-5","type":"journal-article","created":{"date-parts":[[2011,10,27]],"date-time":"2011-10-27T19:34:27Z","timestamp":1319744067000},"page":"371-408","source":"Crossref","is-referenced-by-count":40,"title":["Convex envelopes generated from finitely many compact convex sets"],"prefix":"10.1007","volume":"137","author":[{"given":"Aida","family":"Khajavirad","sequence":"first","affiliation":[]},{"given":"Nikolaos V.","family":"Sahinidis","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2011,10,28]]},"reference":[{"key":"496_CR1","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1287\/moor.8.2.273","volume":"8","author":"F.A. Al-Khayyal","year":"1983","unstructured":"Al-Khayyal F.A., Falk J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273\u2013286 (1983)","journal-title":"Math. Oper. Res."},{"key":"496_CR2","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0167-5060(08)70342-X","volume":"5","author":"E. Balas","year":"1979","unstructured":"Balas E.: Disjunctive programming. Ann. Discrete Math. 5, 3\u201351 (1979)","journal-title":"Ann. Discrete Math."},{"key":"496_CR3","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1137\/0606047","volume":"6","author":"E. Balas","year":"1985","unstructured":"Balas E.: Disjunctive programming and hierarchy of relaxations for discrete optimization problems. SIAM J. Alg. Disc. Meth. 6, 466\u2013486 (1985)","journal-title":"SIAM J. Alg. Disc. Meth."},{"key":"496_CR4","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1080\/10556780902883184","volume":"24","author":"X. Bao","year":"2009","unstructured":"Bao X., Sahinidis N.V., Tawarmalani M.: Multiterm polyhedral relaxations for nonconvex, quadratically-constrained quadratic programs. Optim. Methods Softw. 24, 485\u2013504 (2009)","journal-title":"Optim. Methods Softw."},{"key":"496_CR5","volume-title":"Convex Optimization Theory","author":"D. Bertsekas","year":"2009","unstructured":"Bertsekas D.: Convex Optimization Theory. Athena Scientific, Cambridge (2009)"},{"key":"496_CR6","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1287\/ijoc.15.1.114.15159","volume":"15","author":"M.R. Bussieck","year":"2003","unstructured":"Bussieck M.R., Drud A.S., Meeraus A.: MINLPLib\u2014a collection of test models for mixed-integer nonlinear programming. INFORMS J. Comput. 15, 114\u2013119 (2003)","journal-title":"INFORMS J. Comput."},{"key":"496_CR7","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF03014795","volume":"32","author":"C. Carath\u00e9odory","year":"1911","unstructured":"Carath\u00e9odory C.: Uber den variabilitatsbereich der Fourierschen konstanten von positiven harmonischen funktionen. Rendiconto Circolo Matematico Palermo 32, 193\u2013217 (1911)","journal-title":"Rendiconto Circolo Matematico Palermo"},{"key":"496_CR8","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1007\/s101070050106","volume":"86","author":"S. Ceria","year":"1999","unstructured":"Ceria S., Soares J.: Convex programming for disjunctive convex optimization. Math. Program. 86, 595\u2013614 (1999)","journal-title":"Math. Program."},{"key":"496_CR9","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/s10107-005-0594-3","volume":"106","author":"A. Frangioni","year":"2006","unstructured":"Frangioni A., Gentile C.: Perspective cuts for a class of convex 0\u22121 mixed integer programs. Math. Program. 106, 225\u2013236 (2006)","journal-title":"Math. Program."},{"key":"496_CR10","unstructured":"GLOBAL Library. http:\/\/www.gamsworld.org\/global\/globallib.htm"},{"key":"496_CR11","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1023\/A:1025154322278","volume":"26","author":"I.E. Grossmann","year":"2003","unstructured":"Grossmann I.E., Lee S.: Generalized convex disjunctive programming: nonlinear convex hull relaxation. Comput. Optim. Appl. 26, 83\u2013100 (2003)","journal-title":"Comput. Optim. Appl."},{"key":"496_CR12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/978-3-540-68891-4_1","volume":"5035","author":"O. G\u00fcnl\u00fck","year":"2008","unstructured":"G\u00fcnl\u00fck O., Linderoth J.: Perspective relaxation of mixed integer nonlinear programs with indicator variables. Lect. Notes Comput. Sci. 5035, 1\u201316 (2008)","journal-title":"Lect. Notes Comput. Sci."},{"key":"496_CR13","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-56468-0","volume-title":"Fundamentals of Convex Analysis","author":"J.-B. Hiriart-Urruty","year":"2001","unstructured":"Hiriart-Urruty J.-B., Lemar\u00e9chal C.: Fundamentals of Convex Analysis. Grundlehren Text Editions, New York (2001)"},{"key":"496_CR14","doi-asserted-by":"crossref","first-page":"1451","DOI":"10.1137\/07069359X","volume":"19","author":"M. Jach","year":"2008","unstructured":"Jach M., Michaels D., Weismantel R.: The convex envelope of (n \u2212 1)-convex functions. SIAM J. Optim. 19, 1451\u20131466 (2008)","journal-title":"SIAM J. Optim."},{"key":"496_CR15","doi-asserted-by":"crossref","unstructured":"Khajavirad, A., Sahinidis, N.V.: Convex envelopes of products of convex and component-wise concave functions. J. Global Optim. doi: 10.1007\/s10898-011-9747-5 (2011)","DOI":"10.1007\/s10898-011-9747-5"},{"key":"496_CR16","first-page":"235","volume-title":"Mathematical Programming: The State of the Art","author":"L. Lov\u00e1sz","year":"1982","unstructured":"Lov\u00e1sz L.: Submodular functions and convexity. In: Bachem, A., Grotschel, M., Korte, B. (eds) Mathematical Programming: The State of the Art, pp. 235\u2013257. Springer, Berlin (1982)"},{"key":"496_CR17","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF01580665","volume":"10","author":"G.P. McCormick","year":"1976","unstructured":"McCormick G.P.: Computability of global solutions to factorable nonconvex programs: Part I\u2014Convex underestimating problems. Math. Program. 10, 147\u2013175 (1976)","journal-title":"Math. Program."},{"key":"496_CR18","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1023\/B:JOGO.0000042112.72379.e6","volume":"29","author":"C.A. Meyer","year":"2004","unstructured":"Meyer C.A., Floudas C.A.: Trilinear monomials with mixed sign domains: facets of the convex and concave envelopes. J. Global Optim. 29, 125\u2013155 (2004)","journal-title":"J. Global Optim."},{"key":"496_CR19","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s10107-005-0580-9","volume":"103","author":"C.A. Meyer","year":"2005","unstructured":"Meyer C.A., Floudas C.A.: Convex envelopes for edge-concave functions. Math. Program. 103, 207\u2013224 (2005)","journal-title":"Math. Program."},{"key":"496_CR20","first-page":"131","volume-title":"Gesammelte Abhandlungen von Hermann Minkowski, vol. 2","author":"H. Minkowski","year":"1911","unstructured":"Minkowski H.: Theorie der Knovexen K\u00f6rper, insbesondere begrundung ihres oberfla chenbegriffs. In: Hilbert, D., Speiser, A., Weyl, H. (eds) Gesammelte Abhandlungen von Hermann Minkowski, vol. 2, pp. 131\u2013229. Teubner, Leipsig (1911)"},{"key":"496_CR21","doi-asserted-by":"crossref","unstructured":"Nemhauser, G.L., Wolsey, L.A.: Integer and combinatorial optimization. Wiley Interscience, Series in Discrete Mathematics and Optimization, New York (1988)","DOI":"10.1002\/9781118627372"},{"key":"496_CR22","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1023\/A:1008217604285","volume":"10","author":"A.D. Rikun","year":"1997","unstructured":"Rikun A.D.: A convex envelope formula for multilinear functions. J. Global Optim. 10, 425\u2013437 (1997)","journal-title":"J. Global Optim."},{"key":"496_CR23","volume-title":"Convex Analysis. Princeton Mathematical Series","author":"R.T. Rockafellar","year":"1970","unstructured":"Rockafellar R.T.: Convex Analysis. Princeton Mathematical Series. Princeton University Press, Princeton (1970)"},{"key":"496_CR24","volume-title":"Variational Analysis. A Series of Comprehensive Studies in Mathematics","author":"R.T. Rockafellar","year":"1998","unstructured":"Rockafellar R.T., Wets R.J.-B.: Variational Analysis. A Series of Comprehensive Studies in Mathematics. Springer, Berlin (1998)"},{"key":"496_CR25","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/BF00138693","volume":"8","author":"N.V. Sahinidis","year":"1996","unstructured":"Sahinidis N.V.: BARON: a general purpose global optimization software package. J. Global Optim. 8, 201\u2013205 (1996)","journal-title":"J. Global Optim."},{"key":"496_CR26","first-page":"245","volume":"22","author":"H.D. Sherali","year":"1997","unstructured":"Sherali H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnam. 22, 245\u2013270 (1997)","journal-title":"Acta Math. Vietnam."},{"key":"496_CR27","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1007\/s101070050103","volume":"86","author":"R. Stubbs","year":"1999","unstructured":"Stubbs R., Mehrotra S.: A branch and cut method for 0\u22121 mixed convex programming. Math. Program. 86, 515\u2013532 (1999)","journal-title":"Math. Program."},{"key":"496_CR28","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/s11590-007-0065-2","volume":"2","author":"F. Tardella","year":"2008","unstructured":"Tardella F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2, 363\u2013375 (2008)","journal-title":"Optim. Lett."},{"key":"496_CR29","unstructured":"Tawarmalani, M.: Inclusion certificates and simultaneous convexification of functions. Math. Program. (submitted)"},{"key":"496_CR30","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1007\/s10107-010-0374-6","volume":"124","author":"M. Tawarmalani","year":"2010","unstructured":"Tawarmalani M., Richard J.-P., Chung K.: Strong valid inequalities for orthogonal disjunctions and bilinear covering sets. Math. Program. 124, 481\u2013512 (2010)","journal-title":"Math. Program."},{"key":"496_CR31","unstructured":"Tawarmalani, M., Richard, J.-P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. (submitted)"},{"key":"496_CR32","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1023\/A:1011233805045","volume":"20","author":"M. Tawarmalani","year":"2001","unstructured":"Tawarmalani M., Sahinidis N.V.: Semidefinite relaxations of fractional programs via novel techniques for constructing convex envelopes of nonlinear functions. J. Global Optim. 20, 137\u2013158 (2001)","journal-title":"J. Global Optim."},{"key":"496_CR33","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s10107-002-0308-z","volume":"93","author":"M. Tawarmalani","year":"2002","unstructured":"Tawarmalani M., Sahinidis N.V.: Convex extensions and convex envelopes of l.s.c. functions. Math. Program. 93, 247\u2013263 (2002)","journal-title":"Math. Program."},{"key":"496_CR34","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/s10107-003-0467-6","volume":"99","author":"M. Tawarmalani","year":"2004","unstructured":"Tawarmalani M., Sahinidis N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563\u2013591 (2004)","journal-title":"Math. Program."},{"key":"496_CR35","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/s10107-005-0581-8","volume":"103","author":"M. Tawarmalani","year":"2005","unstructured":"Tawarmalani M., Sahinidis N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225\u2013249 (2005)","journal-title":"Math. Program."},{"key":"496_CR36","doi-asserted-by":"crossref","DOI":"10.1515\/9781400822539","volume-title":"Supermodularity and Complementarity","author":"D.M. Topkis","year":"1998","unstructured":"Topkis D.M.: Supermodularity and Complementarity. Princeton University Press, Princeton (1998)"}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-011-0496-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10107-011-0496-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-011-0496-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T17:08:03Z","timestamp":1593191283000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10107-011-0496-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,28]]},"references-count":36,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2013,2]]}},"alternative-id":["496"],"URL":"https:\/\/doi.org\/10.1007\/s10107-011-0496-5","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,10,28]]}}}