# 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 - 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.

## Citation

Technical report, Department of Management Sciences, University of Iowa, June 2014.