{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T04:25:17Z","timestamp":1768278317250,"version":"3.49.0"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2020,7,13]],"date-time":"2020-07-13T00:00:00Z","timestamp":1594598400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,7,13]],"date-time":"2020-07-13T00:00:00Z","timestamp":1594598400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000084","name":"Directorate for Engineering","doi-asserted-by":"publisher","award":["1727989"],"award-info":[{"award-number":["1727989"]}],"id":[{"id":"10.13039\/100000084","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2021,11]]},"DOI":"10.1007\/s10107-020-01541-x","type":"journal-article","created":{"date-parts":[[2020,7,13]],"date-time":"2020-07-13T16:03:20Z","timestamp":1594656200000},"page":"427-466","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A new framework to relax composite functions in nonlinear programs"],"prefix":"10.1007","volume":"190","author":[{"given":"Taotao","family":"He","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3085-0084","authenticated-orcid":false,"given":"Mohit","family":"Tawarmalani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,13]]},"reference":[{"issue":"2","key":"1541_CR1","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1287\/moor.8.2.273","volume":"8","author":"FA Al-Khayyal","year":"1983","unstructured":"Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273\u2013286 (1983)","journal-title":"Math. Oper. Res."},{"issue":"1\u20132","key":"1541_CR2","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10107-010-0355-9","volume":"124","author":"KM Anstreicher","year":"2010","unstructured":"Anstreicher, K.M., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Math. Program. 124(1\u20132), 33\u201343 (2010)","journal-title":"Math. Program."},{"issue":"1","key":"1541_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12532-014-0073-z","volume":"7","author":"X Bao","year":"2015","unstructured":"Bao, X., Khajavirad, A., Sahinidis, N.V., Tawarmalani, M.: Global optimization of nonconvex problems with multilinear intermediates. Math. Program. Comput. 7(1), 1\u201337 (2015)","journal-title":"Math. Program. Comput."},{"issue":"4\u20135","key":"1541_CR4","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1080\/10556780903087124","volume":"24","author":"P Belotti","year":"2009","unstructured":"Belotti, P., Lee, J., Liberti, L., Margot, F., W\u00e4chter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4\u20135), 597\u2013634 (2009)","journal-title":"Optim. Methods Softw."},{"issue":"4","key":"1541_CR5","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1002\/nav.20011","volume":"51","author":"HP Benson","year":"2004","unstructured":"Benson, H.P.: Concave envelopes of monomial functions over rectangles. Naval Res. Logist. 51(4), 467\u2013476 (2004)","journal-title":"Naval Res. Logist."},{"issue":"2","key":"1541_CR6","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1137\/120878963","volume":"24","author":"D Bienstock","year":"2014","unstructured":"Bienstock, D., Michalka, A.: Cutting-planes for optimization of convex functions over nonconvex sets. SIAM J. Optim. 24(2), 643\u2013677 (2014)","journal-title":"SIAM J. Optim."},{"issue":"1\u20132","key":"1541_CR7","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/s10107-016-1031-5","volume":"162","author":"N Boland","year":"2017","unstructured":"Boland, N., Dey, S.S., Kalinowski, T., Molinaro, M., Rigterink, F.: Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions. Math. Program. 162(1\u20132), 523\u2013535 (2017)","journal-title":"Math. Program."},{"issue":"1\u20132","key":"1541_CR8","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/s10107-016-1045-z","volume":"162","author":"S Burer","year":"2017","unstructured":"Burer, S., K\u0131l\u0131n\u00e7-Karzan, F.: How to convexify the intersection of a second order cone and a nonconvex quadratic. Math. Program. 162(1\u20132), 393\u2013429 (2017)","journal-title":"Math. Program."},{"issue":"4","key":"1541_CR9","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1007\/s10898-009-9484-1","volume":"47","author":"S Cafieri","year":"2010","unstructured":"Cafieri, S., Lee, J., Liberti, L.: On convex relaxations of quadrilinear terms. J. Global Optim. 47(4), 661\u2013685 (2010)","journal-title":"J. Global Optim."},{"issue":"4","key":"1541_CR10","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/BF02712873","volume":"16","author":"TM Chan","year":"1996","unstructured":"Chan, T.M.: Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete Comput. Geom. 16(4), 361\u2013368 (1996)","journal-title":"Discrete Comput. Geom."},{"key":"1541_CR11","unstructured":"CMU-IBM Cyber-Infrastructure for MINLP collaborative site (2019). http:\/\/www.minlp.org. Accessed 10 Oct 2019"},{"key":"1541_CR12","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.disopt.2017.02.001","volume":"25","author":"Y Crama","year":"2017","unstructured":"Crama, Y., Rodr\u00edguez-Heck, E.: A class of valid inequalities for multilinear 0\u20131 optimization problems. Discrete Optim. 25, 28\u201347 (2017)","journal-title":"Discrete Optim."},{"issue":"2","key":"1541_CR13","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1287\/moor.2016.0804","volume":"42","author":"A Del Pia","year":"2016","unstructured":"Del Pia, A., Khajavirad, A.: A polyhedral study of binary polynomial programs. Math. Oper. Res. 42(2), 389\u2013410 (2016)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"1541_CR14","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1137\/16M1095998","volume":"28","author":"A Del Pia","year":"2018","unstructured":"Del Pia, A., Khajavirad, A.: The multilinear polytope for acyclic hypergraphs. SIAM J. Optim. 28(2), 1049\u20131076 (2018)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1541_CR15","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s12532-019-00169-z","volume":"12","author":"A Del Pia","year":"2020","unstructured":"Del Pia, A., Khajavirad, A., Sahinidis, N.V.: On the impact of running intersection inequalities for globally solving polynomial optimization problems. Math. Program. Comput. 12(2), 165\u2013191 (2020)","journal-title":"Math. Program. Comput."},{"key":"1541_CR16","unstructured":"Gleixner, A., Bastubbe, M., Eifler, L., Gally, T., Gamrath, G., Gottwald, R.L., Hendel, G., Hojny, C., Koch, T., L\u00fcbbecke, M.E., Maher, S.J., Miltenberger, M., M\u00fcller, B., Pfetsch, M.E., Puchert, C., Rehfeldt, D., Schl\u00f6sser, F., Schubert, C., Serrano, F., Shinano, Y., Viernickel, J.M., Walter, M., Wegscheider, F., Witt, J.T., Witzig, J.: The SCIP Optimization Suite 6.0. Technical report, Optimization Online (2018)"},{"key":"1541_CR17","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1007\/s10898-016-0450-4","volume":"67","author":"AM Gleixner","year":"2017","unstructured":"Gleixner, A.M., Berthold, T., M\u00fcller, B., Weltge, S.: Three enhancements for optimization-based bound tightening. J. Global Optim. 67, 731\u2013757 (2017)","journal-title":"J. Global Optim."},{"issue":"4","key":"1541_CR18","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/0020-0190(72)90045-2","volume":"1","author":"RL Graham","year":"1972","unstructured":"Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Inf. Process. Lett. 1(4), 132\u2013133 (1972)","journal-title":"Inf. Process. Lett."},{"key":"1541_CR19","volume-title":"Geometric Algorithms and Combinatorial Optimization","author":"M Gr\u00f6tschel","year":"2012","unstructured":"Gr\u00f6tschel, M., Lov\u00e1sz, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2. Springer, Berlin (2012)"},{"issue":"1\u20132","key":"1541_CR20","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s10107-010-0360-z","volume":"124","author":"O G\u00fcnl\u00fck","year":"2010","unstructured":"G\u00fcnl\u00fck, O., Linderoth, J.: Perspective reformulations of mixed integer nonlinear programs with indicator variables. Math. Program. 124(1\u20132), 183\u2013205 (2010)","journal-title":"Math. Program."},{"key":"1541_CR21","doi-asserted-by":"crossref","first-page":"100569","DOI":"10.1016\/j.disopt.2020.100569","volume":"36","author":"A Gupte","year":"2020","unstructured":"Gupte, A., Kalinowski, T., Rigterink, F., Waterer, H.: Extended formulations for convex hulls of some bilinear functions. Discrete Optim. 36, 100569 (2020)","journal-title":"Discrete Optim."},{"key":"1541_CR22","unstructured":"He, T., Tawarmalani, M.: Tractable relaxations of composite functions. Working paper (2018)"},{"key":"1541_CR23","volume-title":"Fundamentals of Convex Analysis","author":"JB Hiriart-Urruty","year":"2012","unstructured":"Hiriart-Urruty, J.B., Lemar\u00e9chal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2012)"},{"key":"1541_CR24","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03199-5","volume-title":"Global Optimization\u2014Deterministic Approaches","author":"R Horst","year":"1996","unstructured":"Horst, R., Tuy, H.: Global Optimization\u2014Deterministic Approaches. Springer, Berlin (1996)"},{"issue":"3","key":"1541_CR25","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-1)-convex functions. SIAM J. Optim. 19(3), 1451\u20131466 (2008)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"1541_CR26","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/s10898-011-9747-5","volume":"52","author":"A Khajavirad","year":"2012","unstructured":"Khajavirad, A., Sahinidis, N.V.: Convex envelopes of products of convex and component-wise concave functions. J. Global Optim. 52(3), 391\u2013409 (2012)","journal-title":"J. Global Optim."},{"issue":"1\u20132","key":"1541_CR27","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s10107-011-0496-5","volume":"137","author":"A Khajavirad","year":"2013","unstructured":"Khajavirad, A., Sahinidis, N.V.: Convex envelopes generated from finitely many compact convex sets. Math. Program. 137(1\u20132), 371\u2013408 (2013)","journal-title":"Math. Program."},{"issue":"1\u20132","key":"1541_CR28","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/s10107-015-0903-4","volume":"154","author":"F K\u0131l\u0131n\u00e7-Karzan","year":"2015","unstructured":"K\u0131l\u0131n\u00e7-Karzan, F., Y\u0131ld\u0131z, S.: Two-term disjunctions on the second-order cone. Math. Program. 154(1\u20132), 463\u2013491 (2015)","journal-title":"Math. Program."},{"issue":"2","key":"1541_CR29","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s10107-012-0606-z","volume":"136","author":"J Luedtke","year":"2012","unstructured":"Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Program. 136(2), 325\u2013351 (2012)","journal-title":"Math. Program."},{"issue":"1","key":"1541_CR30","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF01580665","volume":"10","author":"GP McCormick","year":"1976","unstructured":"McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part i Convex underestimating problems. Math. Program. 10(1), 147\u2013175 (1976)","journal-title":"Math. Program."},{"issue":"2","key":"1541_CR31","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1023\/B:JOGO.0000042112.72379.e6","volume":"29","author":"CA 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(2), 125\u2013155 (2004)","journal-title":"J. Global Optim."},{"issue":"2\u20133","key":"1541_CR32","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/s10898-014-0166-2","volume":"59","author":"R Misener","year":"2014","unstructured":"Misener, R., Floudas, C.A.: ANTIGONE: Algorithms for continuous\/integer global optimization of nonlinear equations. J. Global Optim. 59(2\u20133), 503\u2013526 (2014)","journal-title":"J. Global Optim."},{"issue":"1","key":"1541_CR33","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1080\/10556788.2014.916287","volume":"30","author":"R Misener","year":"2015","unstructured":"Misener, R., Smadbeck, J.B., Floudas, C.A.: Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2. Optim. Methods Softw. 30(1), 215\u2013249 (2015)","journal-title":"Optim. Methods Softw."},{"issue":"1\u20132","key":"1541_CR34","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s10107-016-1084-5","volume":"164","author":"S Modaresi","year":"2017","unstructured":"Modaresi, S., Vielma, J.P.: Convex hull of two quadratic or a conic quadratic and a quadratic inequality. Math. Program. 164(1\u20132), 383\u2013409 (2017)","journal-title":"Math. Program."},{"issue":"2","key":"1541_CR35","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1137\/19M1249825","volume":"30","author":"B Muller","year":"2020","unstructured":"Muller, B., Serrano, F., Gleixner, A.: Using two-dimensional projections for stronger separation and propagation of bilinear terms. SIAM J. Optim. 30(2), 1339\u20131365 (2020)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"1541_CR36","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/s10898-019-00791-0","volume":"75","author":"J Najman","year":"2019","unstructured":"Najman, J., Mitsos, A.: Tighter McCormick relaxations through subgradient propagation. J. Global Optim. 75(3), 565\u2013593 (2019)","journal-title":"J. Global Optim."},{"issue":"2","key":"1541_CR37","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/s10107-017-1138-3","volume":"169","author":"TT Nguyen","year":"2018","unstructured":"Nguyen, T.T., Richard, J.P.P., Tawarmalani, M.: Deriving convex hulls through lifting and projection. Math. Program. 169(2), 377\u2013415 (2018)","journal-title":"Math. Program."},{"issue":"1","key":"1541_CR38","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/BF01589101","volume":"45","author":"M Padberg","year":"1989","unstructured":"Padberg, M.: The boolean quadric polytope: some characteristics, facets and relatives. Math. Program. 45(1), 139\u2013172 (1989)","journal-title":"Math. Program."},{"issue":"4","key":"1541_CR39","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1023\/A:1008217604285","volume":"10","author":"AD Rikun","year":"1997","unstructured":"Rikun, A.D.: A convex envelope formula for multilinear functions. J. Global Optim. 10(4), 425\u2013437 (1997)","journal-title":"J. Global Optim."},{"issue":"1","key":"1541_CR40","first-page":"245","volume":"22","author":"HD Sherali","year":"1997","unstructured":"Sherali, H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta mathematica vietnamica 22(1), 245\u2013270 (1997)","journal-title":"Acta mathematica vietnamica"},{"key":"1541_CR41","volume-title":"A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems","author":"HD Sherali","year":"2013","unstructured":"Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, vol. 31. Springer, Berlin (2013)"},{"issue":"4","key":"1541_CR42","doi-asserted-by":"crossref","first-page":"1230","DOI":"10.1287\/moor.2017.0846","volume":"42","author":"E Speakman","year":"2017","unstructured":"Speakman, E., Lee, J.: Quantifying double McCormick. Math Oper Res 42(4), 1230\u20131253 (2017)","journal-title":"Math Oper Res"},{"key":"1541_CR43","unstructured":"Tawarmalani, M.: Inclusion certificates and simultaneous convexification of functions. Working paper (2010)"},{"issue":"1\u20132","key":"1541_CR44","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.P., Chung, K.: Strong valid inequalities for orthogonal disjunctions and bilinear covering sets. Math. Program. 124(1\u20132), 481\u2013512 (2010)","journal-title":"Math. Program."},{"issue":"1\u20132","key":"1541_CR45","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/s10107-012-0581-4","volume":"138","author":"M Tawarmalani","year":"2013","unstructured":"Tawarmalani, M., Richard, J.P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. 138(1\u20132), 531\u2013577 (2013)","journal-title":"Math. Program."},{"issue":"2","key":"1541_CR46","doi-asserted-by":"crossref","first-page":"133","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 convexification techniques. J. Global Optim. 20(2), 133\u2013154 (2001)","journal-title":"J. Global Optim."},{"issue":"2","key":"1541_CR47","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 envelopes of lower semi-continuous functions. Math. Program. 93(2), 247\u2013263 (2002)","journal-title":"Math. Program."},{"issue":"3","key":"1541_CR48","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(3), 563\u2013591 (2004)","journal-title":"Math. Program."},{"issue":"2","key":"1541_CR49","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(2), 225\u2013249 (2005)","journal-title":"Math. Program."},{"key":"1541_CR50","unstructured":"Vigerske, S.: MINLPLIB 2. In: Proceedings of the XII Global Optimization Workshop MAGO 2014, pp. 137\u2013140 (2014)"}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01541-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-020-01541-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01541-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,10]],"date-time":"2021-10-10T02:50:17Z","timestamp":1633834217000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-020-01541-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,13]]},"references-count":50,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2021,11]]}},"alternative-id":["1541"],"URL":"https:\/\/doi.org\/10.1007\/s10107-020-01541-x","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,13]]},"assertion":[{"value":"27 August 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 July 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}