How to convexify the intersection of a second order cone and a nonconvex quadratic

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−xj)(xj−u) ≤ 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∩Q and K∩Q∩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 assumptions, we derive a simple, computable convex relaxations K ∩ S and K ∩ S ∩ H, where S is a SOCr cone. Under further assumptions, 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.