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, (mathcal {E}), and a split disjunction, ((l - x_j)(x_j - u) le 0) with (l < u), equals the intersection of (mathcal {E}) with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form (mathcal {K}cap mathcal {Q}) and (mathcal {K}cap mathcal {Q}cap H), where (mathcal {K}) is a SOCr cone, (mathcal {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 simple, computable convex relaxations (mathcal {K}cap mathcal {S}) and (mathcal {K}cap mathcal {S}cap H), where (mathcal {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.