  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, Kilinc, and Vielma, and by Andersen and Jensen.
