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A Conic Representation of the Convex Hull of Disjunctive Sets and Conic Cuts for Integer Second Order Cone Optimization

  • Pietro Belotti
  • Julio C. Góez
  • Imre Pólik
  • Ted K. Ralphs
  • Tamás Terlaky
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 134)

Abstract

We study the convex hull of the intersection of a convex set E and a disjunctive set. This intersection is at the core of solution techniques for Mixed Integer Convex Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction as E, then the convex hull is the intersection of E with K (resp., C).The existence of such a cone (resp., a cylinder) is difficult to prove for general conic optimization. We prove existence and unicity of a second order cone (resp., a cylinder), when E is the intersection of an affine space and a second order cone (resp., a cylinder). We also provide a method for finding that cone, and hence the convex hull, for the continuous relaxation of the feasible set of a Mixed Integer Second Order Cone Optimization (MISOCO) problem, assumed to be the intersection of an ellipsoid with a general linear disjunction. This cone provides a new conic cut for MISOCO that can be used in branch-and-cut algorithms for MISOCO problems.

Keywords

Conic cuts Mixed integer optimization Second order cone optimization 

Subject Classification:

90C10 90C11 90C20 

Notes

Acknowledgements

The authors Pietro Belotti, Imre Pólik and Tamás Terlaky acknowledge the support of Lehigh University with a start up package for the development of this research. The authors Julio C. Góez and Tamás Terlaky acknowledge the support of the Airforce Research Office grant # FA9550-10-1-0404 for the development of this research.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Pietro Belotti
    • 1
  • Julio C. Góez
    • 2
  • Imre Pólik
    • 3
  • Ted K. Ralphs
    • 4
  • Tamás Terlaky
    • 4
  1. 1.Xpress Optimization TeamFICOBirminghamUK
  2. 2.GERAD and École Polytechnique de MontréalMontrealCanada
  3. 3.SAS InstituteCaryUSA
  4. 4.Department of Industrial and Systems EngineeringLehigh UniversityBethlehemUSA

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