A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown---by several authors using different techniques---that the convex hull of the intersection of an ellipsoid, $\E$, and a split disjunction, $(l - x_j)(x_j - u) \le 0$ with $l < u$, equals the intersection of $\E$ with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form $\K \cap \Q$ and $\K \cap \Q \cap H$, where $\K$ is a SOCr cone, $\Q$ is a nonconvex cone defined by a single homogeneous quadratic, and $H$ is an affine hyperplane. Under several easy-to-verify conditions, we derive a simple, computable convex relaxations $\K \cap \S$ and $\K \cap \S \cap H$, where $\S$ is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.
Technical report, Department of Management Sciences, University of Iowa, June 2014.