Mathematical Programming

, Volume 154, Issue 1–2, pp 463–491

Two-term disjunctions on the second-order cone

Full Length Paper Series B

DOI: 10.1007/s10107-015-0903-4

Cite this article as:
Kılınç-Karzan, F. & Yıldız, S. Math. Program. (2015) 154: 463. doi:10.1007/s10107-015-0903-4


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 study the structure of the convex hull of a two-term disjunction applied to the second-order cone and develop a methodology to derive closed-form expressions for convex inequalities describing the resulting convex hull. 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. In particular, 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.


Mixed-integer conic programming Second-order cone programming Cutting planes Disjunctive cuts 

Mathematics Subject Classification

90C11 90C26 

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  1. 1.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA

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