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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.

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