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 un-dominated 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, Kilinc, and Vielma, and by Andersen and Jensen.
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