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

Abstract

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.