Small and strong formulations for unions of convex sets from the Cayley embedding

  • Juan Pablo Vielma
Full Length Paper Series A


There is often a significant trade-off between formulation strength and size in mixed integer programming. When modeling convex disjunctive constraints (e.g. unions of convex sets), adding auxiliary continuous variables can sometimes help resolve this trade-off. However, standard formulations that use such auxiliary continuous variables can have a worse-than-expected computational effectiveness, which is often attributed precisely to these auxiliary continuous variables. For this reason, there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables. We introduce a technique to construct formulations without these detrimental continuous auxiliary variables. To develop this technique we introduce a natural non-polyhedral generalization of the Cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding. We then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints. In particular, we show it can recover and generalize all known strong formulations without continuous auxiliary variables.


Mixed integer nonlinear programming Mixed integer programming formulations Disjunctive constraints 

Mathematics Subject Classification

90C11 90C25 90C30 



This research was partially supported by NSF under Grant CMMI-1351619. We thank two anonymous referees for their constructive comments that improved the paper’s presentation.


  1. 1.
    Andradas, C., Ruiz, J.M.: Ubiquity of łojasiewicz’s example of a nonbasic semialgebraic set. Mich. Math. J. 41, 465–472 (1994)CrossRefzbMATHGoogle Scholar
  2. 2.
    Balas, E.: On the convex-hull of the union of certain polyhedra. Oper. Res. Lett. 7, 279–283 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. Society for Industrial Mathematics (2001)Google Scholar
  4. 4.
    Bestuzheva, K., Hijazi, H., Coffrin, C.: Convex Relaxations for Quadratic On/Off Constraints and Applications to Optimal Transmission Switching (2016). Optimization Online.
  5. 5.
    Blair, C.: Representation for multiple right-hand sides. Math. Program. 49, 1–5 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blekherman, G., Parrilo, P., Thomas, R.: Semidefinite Optimization and Convex Algebraic Geometry. MPS-SIAM Series on Optimization. SIAM (2013)Google Scholar
  7. 7.
    Bonami, P., Lodi, A., Tramontani, A., Wiese, S.: On mathematical programming with indicator constraints. Math. Program. 151, 191–223 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. 86, 595–614 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Günlük, O., Linderoth, J.: Perspective reformulations of mixed integer nonlinear programs with indicator variables. Math. Program. 124, 183–205 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Helton, J.W., Nie, J.: Sufficient and necessary conditions for semidefinite representability of convex hulls and sets. SIAM J. Optim. 20, 759–791 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hijazi, H., Bonami, P., Cornuéjols, G., Ouorou, A.: Mixed-integer nonlinear programs featuring “on/off” constraints. Comput. Optim. Appl. 52, 537–558 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hijazi, H., Bonami, P., Ouorou, A.: A Note on Linear On/Off Constraints (2014). Optimization Online.
  13. 13.
    Hijazi, H., Coffrin, C., Van Hentenryck, P.: Convex quadratic relaxations for mixed-integer nonlinear programs in power systems. Math. Program. Comput. 9, 321–367 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Heidelberg (2001)CrossRefzbMATHGoogle Scholar
  15. 15.
    Huber, B., Rambau, J., Santos, F.: The cayley trick, lifting subdivisions and the bohne-dress theorem on zonotopal tilings. J. Eur. Math. Soc. 2, 179–198 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jeroslow, R.G.: A simplification for some disjunctive formulations. Eur. J. Oper. Res. 36, 116–121 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Karavelas, M.I., Konaxis, C., Tzanaki, E.: The maximum number of faces of the Minkowski sum of three convex polytopes. In: da Fonseca, G.D., Lewiner, T., Peñaranda, L.M., Chan, T.M., Klein, R. (eds.) Symposuim on Computational Geometry 2013, SoCG’13, Rio de Janeiro, Brazil, June 17–20, 2013, pp. 187–196. ACM (2013)Google Scholar
  18. 18.
    Lubin, M., Yamangil, E., Bent, R., Vielma, J.P.: Extended formulations in mixed-integer convex programming. In: Louveaux, Q., Skutella, M. (eds.) Integer Programming and Combinatorial Optimization—18th International Conference, IPCO 2016, Liège, Belgium, June 1–3, 2016, Proceedings, LNCS, vol. 9682, pp. 102–113. Springer, Berlin (2016)Google Scholar
  19. 19.
    Lubin, M., Yamangil, E., Bent, R., Vielma, J.P.: Polyhedral approximation in mixed-integer convex optimization. Math. Program. (to appear) (2017).
  20. 20.
    Rockafellar, R.: Convex Analysis. Princeton Landmarks in Mathematics and Physics. Princeton University Press, Princeton (2015)Google Scholar
  21. 21.
    Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0–1 mixed convex programming. Math. Program. 86, 515–532 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tawarmalani, M.: Inclusion Certificates and Simultaneous Convexification of Functions (2010). Optimization Online.
  23. 23.
    Tawarmalani, M., Richard, J., Chung, K.: Strong valid inequalities for orthogonal disjunctions and bilinear covering sets. Math. Program. 124, 481–512 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Vielma, J.P.: Embedding formulations and complexity for unions of polyhedra. Manag. Sci. (to appear) (2017).
  25. 25.
    Vielma, J.P.: Mixed integer linear programming formulation techniques. SIAM Rev. 57, 3–57 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vielma, J.P., Nemhauser, G.L.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Program. 128, 49–72 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Weibel, C.: Minkowski Sums of Polytopes: Combinatorics and Computation. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2007)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Sloan School of ManagementMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations