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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
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