Abstract
Balas introduced intersection cuts for mixed integer linear sets. Intersection cuts are given by closed form formulas and form an important class of cuts for solving mixed integer linear programs. In this paper we introduce an extension of intersection cuts to mixed integer conic quadratic sets. We identify the formula for the conic quadratic intersection cut by formulating a system of polynomial equations with additional variables that are satisfied by points on a certain piece of the boundary defined by the intersection cut. Using a software package from algebraic geometry we then eliminate variables from the system and get a formula for the intersection cut in dimension three. This formula is finally generalized and proved for any dimension. The intersection cut we present generalizes a conic quadratic cut introduced by Modaresi, Kilinc and Vielma.
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References
Andersen, K., Cornuéjols, G., Li, Y.: Split closure and intersection cuts. Mathematical Programming A 102, 457–493 (2005)
Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Mathematical Programming 122(1), 1–20 (2010)
Atamtürk, A., Narayanan, V.: Lifting for conic mixed-integer programming. Mathematical Programming A 126, 351–363 (2011)
Balas, E.: Intersection cuts - a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)
Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Mathematical Programming 58, 295–324 (1993)
Bixby, R.E., Gu, Z., Rothberg, E., Wunderling, R.: Mixed integer programming: A progress report. In: Grötschel, M. (ed.) The Sharpest Cut: The Impact of Manfred Padberg and his Work. MPS/SIAM Ser. Optim., pp. 309–326 (2004)
Çezik, M., Iyengar, G.: Cuts for mixed 0-1 conic programming. Mathematical Programming 104(1), 179–202 (2005)
Cook, W.J., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Mathematical Programming 47, 155–174 (1990)
Gomory, R.: An algorithm for the mixed integer problem. Technical Report RM-2597, The Rand Coporation (1960)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-3 — A computer algebra system for polynomial computations, University of Kaiserslautern (2011), http://www.singular.uni-kl.de
Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra, 2nd edn. Springer Publishing Company, Incorporated (2007)
Modaresi, S., Kilinc, M., Vielma, J.P.: Split Cuts for Conic Programming. Poster presented at the MIP 2012 Workshop at UC Davis
Nemhauser, G., Wolsey, L.: A recursive procedure to generate all cuts for 0-1 mixed integer programs. Mathematical Programming 46, 379–390 (1990)
Ranestad, K., Sturmfels, B.: The convex hull of a variety. In: Branden, P., Passare, M., Putinar, M. (eds.) Notions of Positivity and the Geometry of Polynomials. Trends in Mathematics, pp. 331–344. Springer, Basel (2011)
Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0-1 mixed convex programming. Mathematical Programming 86(3), 515–532 (1999)
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Andersen, K., Jensen, A.N. (2013). Intersection Cuts for Mixed Integer Conic Quadratic Sets. In: Goemans, M., Correa, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2013. Lecture Notes in Computer Science, vol 7801. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36694-9_4
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DOI: https://doi.org/10.1007/978-3-642-36694-9_4
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