Abstract
Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we develop a methodology to derive closed-form expressions for inequalities describing the convex hull of a two-term disjunction applied to the second-order cone. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly nonlinear, valid inequalities that correspond to these linear inequalities. We identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull. Our results on two-term disjunctions on the second-order cone generalize related results on split cuts by Modaresi, Kılınç, and Vielma, and by Andersen and Jensen.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Benson, H., Saglam, U.: Mixed-integer second-order cone programming: A survey. Tutorials in Operations Research. In: INFORMS, Hanover, MD, pp. 13–36 (2013)
Júdice, J., Sherali, H., Ribeiro, I., Faustino, A.: A complementarity-based partitioning and disjunctive cut algorithm for mathematical programming problems with equilibrium constraints. Journal of Global Optimization 136, 89–114 (2006)
Belotti, P.: Disjunctive cuts for nonconvex MINLP. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol. 154. Springer, New York (2012)
Bonami, P., Conforti, M., Cornuéjols, G., Molinaro, M., Zambelli, G.: Cutting planes from two-term disjunctions. Operations Research Letters 41, 442–444 (2013)
Balas, E.: Intersection cuts - a new type of cutting planes for integer programming. Operations Research 19, 19–39 (1971)
Çezik, M., Iyengar, G.: Cuts for mixed 0-1 conic programming. Mathematical Programming 104(1), 179–202 (2005)
Stubbs, R., Mehrotra, S.: A branch-and-cut method for 0-1 mixed convex programming. Mathematical Programming 86(3), 515–532 (1999)
Drewes, S.: Mixed Integer Second Order Cone Programming. PhD thesis, Technische Universität Darmstadt (2009)
Bonami, P.: Lift-and-project cuts for mixed integer convex programs. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 52–64. Springer, Heidelberg (2011)
Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Mathematical Programming 122(1), 1–20 (2010)
Kılınç-Karzan, F.: On minimal valid inequalities for mixed integer conic programs. GSIA Working Paper Number: 2013-E20, GSIA, Carnegie Mellon University, Pittsburgh, PA (June 2013)
Bienstock, D., Michalka, A.: Cutting planes for optimization of convex functions over nonconvex sets. Working paper (May 2013)
Dadush, D., Dey, S., Vielma, J.: The split closure of a strictly convex body. Operations Research Letters 39, 121–126 (2011)
Andersen, K., Jensen, A.: Intersection cuts for mixed integer conic quadratic sets. In: Goemans, M., Correa, J. (eds.) IPCO 2013. LNCS, vol. 7801, pp. 37–48. Springer, Heidelberg (2013)
Modaresi, S., Kılınç, M., Vielma, J.: Intersection cuts for nonlinear integer programming: Convexification techniques for structured sets. Working paper (March 2013)
Belotti, P., Goez, J.C., Polik, I., Ralphs, T., Terlaky, T.: A conic representation of the convex hull of disjunctive sets and conic cuts for integer second order cone optimization. Technical report, Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA (June 2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Kılınç-Karzan, F., Yıldız, S. (2014). Two-Term Disjunctions on the Second-Order Cone. In: Lee, J., Vygen, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2014. Lecture Notes in Computer Science, vol 8494. Springer, Cham. https://doi.org/10.1007/978-3-319-07557-0_29
Download citation
DOI: https://doi.org/10.1007/978-3-319-07557-0_29
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07556-3
Online ISBN: 978-3-319-07557-0
eBook Packages: Computer ScienceComputer Science (R0)