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 TerlakyEmail author
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 134)


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.


Conic cuts Mixed integer optimization Second order cone optimization 

Subject Classification:

90C10 90C11 90C20 



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.


  1. 1.
    Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Math. Program. 122(1), 1–20 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Atamtürk, A., Narayanan, V.: Lifting for conic mixed-integer programming. Math. Program. A 126, 351–363 (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Atamtürk, A., Berenguer, G., Shen, Z.J.: A conic integer programming approach to stochastic joint location-inventory problems. Oper. Res. 60(2), 366–381 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Balas, E.: Disjunctive programming. In: Hammer, P.L., Johnson, E.L., Korte, B.H. (eds.) Annals of Discrete Mathematics 5: Discrete Optimization, pp. 3–51. North Holland, Amsterdam, The Netherlands (1979)Google Scholar
  5. 5.
    Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discret. Appl. Math. 89(1–3), 3–44 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)CrossRefzbMATHGoogle Scholar
  7. 7.
    Barvinok, A.: A Course in Convexity. American Mathematical Society, Providence, RI (2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    Belotti, P., Góez, J., Pólik, I., Ralphs, T., Terlaky, T.: On families of quadratic surfaces having fixed intersections with two hyperplanes. Discret. Appl. Math. 161(16–17), 2778–2793 (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Ben-Tal, A., Nemirovski, A.: On polyhedral approximations of the second-order cone. Math. Oper. Res. 26(2), 193–205 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Bertsimas, D., Shioda, R.: Algorithm for cardinality-constrained quadratic optimization. Comput. Optim. Appl. 43(1), 1–22 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Çezik, M., Iyengar, G.: Cuts for mixed 0–1 conic programming. Math. Program. 104(1), 179–202 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Cornuéjols, G.: Valid inequalities for mixed integer linear programs. Math. Program. 112(1), 3–44 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Cornuéjols, G., Lemaréchal, C.: A convex-analysis perspective on disjunctive cuts. Math. Program. 106(2), 567–586 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Dadush, D., Dey, S., Vielma, J.: The split closure of a strictly convex body. Oper. Res. Lett. 39(2), 121–126 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Drewes, S.: Mixed integer second order cone programming. Ph.D. thesis, Technische Universität Darmstadt, Germany (2009)Google Scholar
  16. 16.
    Fampa, M., Maculan, N.: Using a conic formulation for finding Steiner minimal trees. Numer. Algorithms 35(2), 315–330 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Júdice, J.J., Sherali, H.D., Ribeiro, I.M., Faustino, A.M.: A complementarity-based partitioning and disjunctive cut algorithm for mathematical programming problems with equilibrium constraints. J. Glob. Optim. 36, 89–114 (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Krokhmal, P.A., Soberanis, P.: Risk optimization with p-order conic constraints: a linear programming approach. Eur. J. Oper. Res. 201(3), 653–671 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Kumar, M., Torr, P., Zisserman, A.: Solving Markov random fields using second order cone programming relaxations. In: 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 1045–1052 (2006)Google Scholar
  21. 21.
    Masihabadi, S., Sanjeevi, S., Kianfar, K.: n-Step conic mixed integer rounding inequalities. Optimization Online (2011).
  22. 22.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1999)zbMATHGoogle Scholar
  23. 23.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  24. 24.
    Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0–1 mixed convex programming. Math. Program. 86(3), 515–532 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Vielma, J., Ahmed, S., Nemhauser, G.: A lifted linear programming branch-and-bound algorithm for mixed-integer conic quadratic programs. INFORMS J. Comput. 20(3), 438–450 (2008)CrossRefMathSciNetzbMATHGoogle Scholar

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
    Email author
  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

Personalised recommendations