Mathematical Programming

, Volume 162, Issue 1–2, pp 393–429 | Cite as

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

Full Length Paper Series A


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.


Convex hull Disjunctive programming Mixed-integer linear programming Mixed-integer nonlinear programming Mixed-integer quadratic programming Nonconvex quadratic programming Second-order-cone programming Trust-region subproblem 

Mathematics Subject Classification

90C25 90C10 90C11 90C20 90C26 

Supplementary material

10107_2016_1045_MOESM1_ESM.pdf (904 kb)
Supplementary material 1 (pdf 903 KB)


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of IowaIowa CityUSA
  2. 2.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA

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