How to convexify the intersection of a second order cone and a nonconvex quadratic. (English) Zbl 1358.90095

Summary: 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, \((l - x_j)(x_j - u) \leq 0\) 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.


90C25 Convex programming
90C10 Integer programming
90C11 Mixed integer programming
90C20 Quadratic programming
90C26 Nonconvex programming, global optimization


Full Text: DOI arXiv


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