Mixed-Integer Convex Representability

References

• [1] Abhishek K, Leyffer S, Linderoth J (2010) FilMINT: An outer approximation-based solver for convex mixed-integer nonlinear programs. INFORMS J. Comput. 22(4):555–567.Link, Google Scholar
• [2] Achterberg T, Wunderling R (2013) Mixed integer programming: Analyzing 12 years of progress. Jünger M, Reinelt G, eds. Facets of Combinatorial Optimization: Festschrift for Martin Grötschel (Springer, Berlin, Heidelberg), 449–481.Google Scholar
• [3] Alizadeh F, Goldfarb D (2003) Second-order cone programming. Math. Programming 95(1):3–51.Crossref, Google Scholar
• [4] Barvinok A (2002) A Course in Convexity, vol. 54 (American Mathematical Society, Providence, RI).Crossref, Google Scholar
• [5] Basu A, Conforti M, Cornuéjols G, Zambelli G (2010) Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35(3):704–720.Link, Google Scholar
• [6] Basu A, Martin K, Ryan CT, Wang G (2019) Mixed-integer linear representability, disjunctions, and Chvátal functions—Modeling implications. Math. Oper. Res. 44(4):1264–1285.Link, Google Scholar
• [7] Benson HY, Sağlam Ü (2013) Mixed-integer second-order cone programming: A survey. Theory Driven by Influential Applications, TutORials in Operations Research (INFORMS, Catonsville, MD), 13–36.Google Scholar
• [8] Ben-Tal A, Nemirovski A (2001) Lectures on Modern Convex Optimization (Society for Industrial and Applied Mathematics, Philadelphia).Crossref, Google Scholar
• [9] Bixby RE (2012) A brief history of linear and mixed-integer programming computation. Documenta Math. Optimization Stories, 107–121.Google Scholar
• [10] Bonami P, Biegler LT, Conn AR, Cornuéjols G, Grossmann IE, Laird CD, Lee J, et al. (2008) An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim. 5(2):186–204.Crossref, Google Scholar
• [11] Borwein J, Lewis AS (2010) Convex Analysis and Nonlinear Optimization: Theory and Examples, CMS Books in Mathematics (Springer, New York).Google Scholar
• [12] Boyd S, Vandenberghe L (2004) Convex Optimization (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [13] Ceria S, Soares J (1999) Convex programming for disjunctive convex optimization. Math. Programming 86(3):595–614.Crossref, Google Scholar
• [14] Coey C, Lubin M, Vielma JP (2020) Outer approximation with conic certificates for mixed-integer convex problems. Math. Programming Comput. 12:249–293.Crossref, Google Scholar
• [15] Conforti M, Cornuéjols G, Zambelli G (2014) Integer Programming, Graduate Texts in Mathematics (Springer International Publishing, Heidelberg).Crossref, Google Scholar
• [16] Dadush D, Dey SS, Vielma JP (2011) The Chvátal-Gomory closure of a strictly convex body. Math. Oper. Res. 36(2):227–239.Link, Google Scholar
• [17] Dadush D, Dey SS, Vielma JP (2014) On the Chvátal-Gomory closure of a compact convex set. Math. Programming 145:327–348.Crossref, Google Scholar
• [18] Del Pia A, Poskin J (2016) On the mixed binary representability of ellipsoidal regions. Louveaux Q, Skutella M, eds. Integer Programming Combin. Optim. 18th Internat. Conf. Proc. Lecture Notes Comput. Sci., vol. 9682 (Springer, Berlin, Heidelberg), 214–225.Google Scholar
• [19] Del Pia A, Poskin J (2018) Ellipsoidal mixed-integer representability. Math. Programming 172(1):351–369.Crossref, Google Scholar
• [20] Del Pia A, Poskin J (2019) Characterizations of mixed binary convex quadratic representable sets. Math. Programming 177(1):371–394.Crossref, Google Scholar
• [21] Del Pia A, Dey SS, Molinaro M (2017) Mixed-integer quadratic programming is in NP. Math. Programming 162(1–2):225–240.Crossref, Google Scholar
• [22] Dey SS, Morán DA (2013) Some properties of convex hulls of integer points contained in general convex sets. Math. Programming 141(1):507–526.Crossref, Google Scholar
• [23] Duran MA, Grossmann IE (1986) An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Programming 36(3):307–339.Crossref, Google Scholar
• [24] Fawzi H, Gouveia J, Parrilo PA, Saunderson J, Thomas RR (2020) Lifting for Simplicity: Concise descriptions of convex sets. Preprint, submitted February 22, https://arxiv.org/abs/2002.09788.Google Scholar
• [25] Gally T, Pfetsch ME, Ulbrich S (2018) A framework for solving mixed-integer semidefinite programs. Optim. Methods Software 33(3):594–632.Crossref, Google Scholar
• [26] Hardy GH, Wright EM, Wright EW (1979) An Introduction to the Theory of Numbers, Oxford Science Publications (Clarendon Press, Oxford, UK).Google Scholar
• [27] Hemmecke R, Weismantel R (2007) Representation of sets of lattice points. SIAM J. Optim. 18(1):133–137.Crossref, Google Scholar
• [28] Hiriart-Urruty J-B, Lemaréchal C (1996) Convex Analysis and Minimization Algorithms (Springer Verlag, Heidelberg).Google Scholar
• [29] Hiriart-Urruty J-B, Lemaréchal C (2001) Fundamentals of Convex Analysis (Springer Verlag, Heidelberg).Crossref, Google Scholar
• [30] Huchette J, Vielma JP (2019) A combinatorial approach for small and strong formulations of disjunctive constraints. Math. Oper. Res. 44(3):793–820.Link, Google Scholar
• [31] Jeroslow RG (1987) Representability in mixed integer programming, I: Characterization results. Discrete Appl. Math. 17(3):223–243.Crossref, Google Scholar
• [32] Jeroslow RG, Lowe JK (1984) Modeling with integer variables. Math. Programming Stud. 22:167–184.Crossref, Google Scholar
• [33] Jones JP, Sato D, Wada H, Wiens D (1976) Diophantine representation of the set of prime numbers. Amer. Math. Monthly 83(6):449–464.Crossref, Google Scholar
• [34]Jünger M, Liebling TM, Naddef D, Nemhauser GL, Pulleyblank WR, Reinelt G, Rinaldi G, Wolsey LA, eds. (2010) 50 Years of Integer Programming 1958-2008—From the Early Years to the State-of-the-Art (Springer, Berlin, Heidelberg).Crossref, Google Scholar
• [35] Köppe M (2012) On the complexity of nonlinear mixed-integer optimization. Lee J, Leyffer S, eds. Mixed integer nonlinear programming. The IMA Volumes in Mathematics and its Applications, vol. 154 (Springer-Verlag, New York), 533–557.Google Scholar
• [36] Liu M, Pan Z, Xu K, Manocha D (2020) New formulation of mixed-integer conic programming for globally optimal grasp planning. IEEE Robotics Automation Lett. 5(3):4663–4670.Google Scholar
• [37]Lovász, L (1989) Geometry of numbers and integer programming. Iri M, Tanabe K, eds. Mathematical Programming: Recent Developments and Applications, vol. 6 (Springer, Netherlands), 177–210.Google Scholar
• [38] Lubin M, Zadik I, Vielma JP (2017) Mixed-integer convex representability. Eisenbrand F, Könemann J, eds. Integer Programming Combin. Optim. 19th Internat. Conf. Proc. Lecture Notes Comput. Sci., vol. 10328 (Springer, Berlin, Heidelberg), 392–404.Google Scholar
• [39] Lubin M, Yamangil E, Bent R, Vielma JP (2016) Extended formulations in mixed-integer convex programming. Louveaux Q, Skutella M, eds. Integer Programming Combin. Optim. 18th Internat. Conf. Proc. Lecture Notes Comput. Sci., vol. 9682 (Springer, Berlin, Heidelberg), 102–113.Google Scholar
• [40] Lubin M, Yamangil E, Bent R, Vielma JP (2018) Polyhedral approximation in mixed-integer convex optimization. Math. Programming 172(1):139–168.Crossref, Google Scholar
• [41] Morán DA, Dey SS (2011) On maximal S-free convex sets. SIAM J. Discrete Math. 25(1):379–393.Crossref, Google Scholar
• [42] Morán DA, Dey SS (2016) Closedness of integer hulls of simple conic sets. SIAM J. Discrete Math. 30(1):70–99.Crossref, Google Scholar
• [43] Morán D, Dey SS, Vielma JP (2012) Strong dual for conic mixed-integer programs. SIAM J. Optim. 22(3):1136–1150.Crossref, Google Scholar
• [44] MOSEK ApS (2018) MOSEK modeling cookbook 3.0. Accessed June 11, 2021, https://docs.mosek.com/MOSEKModelingCookbook-letter.pdf.Google Scholar
• [45] Ni SX, So AM (2018) Mixed-integer semidefinite relaxation of joint admission control and beamforming: An soc-based outer approximation approach with provable guarantees. 2018 IEEE 19th Internat. Workshop Signal Processing Adv. Wireless Comm., 1–5.Google Scholar
• [46] Nugroho SA, Taha AF, Gatsis N, Summers TH, Krishnan R (2019) Algorithms for joint sensor and control nodes selection in dynamic networks. Automatica J. IFAC. 106:124–133.Crossref, Google Scholar
• [47] Rockafellar RT (1997) Convex Analysis, Princeton Landmarks in Mathematics and Physics (Princeton University Press, NJ).Google Scholar
• [48] Rockafellar RT, Wets M, Wets RJB (2009) Variational Analysis, Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, Heidelberg).Google Scholar
• [49] Schrijver A (1998) Theory of Linear and Integer Programming (John Wiley & Sons, West Sussex, UK).Google Scholar
• [50] Stubbs RA, Mehrotra S (1999) A branch-and-cut method for 0-1 mixed convex programming. Math. Programming 86(3):515–532.Crossref, Google Scholar
• [51] Vielma JP (2015) Mixed integer linear programming formulation techniques. SIAM Rev. 57(1):3–57.Crossref, Google Scholar
• [52] Vielma JP (2019) Small and strong formulations for unions of convex sets from the Cayley embedding. Math. Programming 177(1–2):21–53.Crossref, Google Scholar