Mathematical Programming

, Volume 159, Issue 1–2, pp 585–605 | Cite as

On sublinear inequalities for mixed integer conic programs

Short Communication Series A

Abstract

This paper studies \(\mathcal {K}\)-sublinear inequalities, a class of inequalities with strong relations to \(\mathcal {K}\)-minimal inequalities for disjunctive conic sets. We establish a stronger result on the sufficiency of \(\mathcal {K}\)-sublinear inequalities. That is, we show that when \(\mathcal {K}\) is the nonnegative orthant or the second-order cone, \(\mathcal {K}\)-sublinear inequalities together with the original conic constraint are always sufficient for the closed convex hull description of the associated disjunctive conic set. When \(\mathcal {K}\) is the nonnegative orthant, \(\mathcal {K}\)-sublinear inequalities are tightly connected to functions that generate cuts—so called cut-generating functions. In particular, we introduce the concept of relaxed cut-generating functions and show that each \({\mathbb {R}}^n_+\)-sublinear inequality is generated by one of these. We then relate the relaxed cut-generating functions to the usual ones studied in the literature. Recently, under a structural assumption, Cornuéjols, Wolsey and Yıldız established the sufficiency of cut-generating functions in terms of generating all nontrivial valid inequalities of disjunctive sets where the underlying cone is nonnegative orthant. We provide an alternate and straightforward proof of this result under the same assumption as a consequence of the sufficiency of \(\mathbb {R}^n_+\)-sublinear inequalities and their connection with relaxed cut-generating functions.

Keywords

Valid inequalities Sublinear inequalities Cut-generating functions Mixed integer conic programming 

Mathematics Subject Classification

Primary 90C11 Secondary 52A41 90C26 

References

  1. 1.
    Andersen, K., Jensen, A.N.: Intersection cuts for mixed integer conic quadratic sets. In: Proceedings of IPCO 2013, Volume 7801 of Lecture Notes in Computer Science, pp. 37–48. Valparaiso, Chile (2013)Google Scholar
  2. 2.
    Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Math. Program. 122(1), 1–20 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bachem, A., Johnson, E.L., Schrader, R.: A characterization of minimal valid inequalities for mixed integer programs. Oper. Res. Lett. 1, 63–66 (1982)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bachem, A., Schrader, R.: Minimal inequalities and subadditive duality. SIAM J. Control Optim. 18, 437–443 (1980)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Minimal inequalities for an infinite relaxation of integer programs. SIAM J. Discrete Math. 24(1), 158–168 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Belotti, P., Góez, J.C., Pólik, I., Ralphs, T.K., Terlaky, T.: On families of quadratic surfaces having fixed intersections with two hyperplanes. Discrete Appl. Math. 161(16), 2778–2793 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Belotti, P., Góez, J.C., Pólik, I., Ralphs, T.K., Terlaky, T.: A conic representation of the convex hull of disjunctive sets and conic cuts for integer second order cone optimization. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds.) Numerical Analysis and Optimization, Volume 134 of Springer Proceedings in Mathematics & Statistics, pp. 1–35. Springer International Publishing, Switzerland (2015)Google Scholar
  8. 8.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization. SIAM, Philadehia (2001)CrossRefMATHGoogle Scholar
  9. 9.
    Bienstock, D., Michalka, A.: Cutting-planes for optimization of convex functions over nonconvex sets. SIAM J. Optim. 24(2), 643–677 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Blair, C.E.: Minimal inequalities for mixed integer programs. Discrete Math. 24, 147–151 (1978)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34(3), 538–546 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Burer, S., Kılınç-Karzan, F.: How to convexify the intersection of a second order cone and a nonconvex quadratic. Technical report, June 2014. Revised June (2015). http://www.andrew.cmu.edu/user/fkilinc/files/nonconvex_quadratics.pdf
  13. 13.
    Conforti, M., Cornuéjols, G., Daniilidis, A., Lemaréchal, C., Malick, J.: Cut-generating functions and \(s\)-free sets. Math. Oper. Res. 40(2), 276–301 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cornuéjols, G., Wolsey, L., Yıldız, S.: Sufficiency of cut-generating functions. Math. Program. 152, 643–651 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dadush, D., Dey, S.S., Vielma, J.P.: The split closure of a strictly convex body. Oper. Res. Lett. 39, 121–126 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra. Math. Program. 3, 23–85 (1972)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jeroslow, R.G.: Cutting plane theory: algebraic methods. Discrete Math. 23, 121–150 (1978)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jeroslow, R.G.: Minimal inequalities. Math. Program. 17, 1–15 (1979)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Johnson, E.L.: On the group problem for mixed integer programming. Math. Program. 2, 137–179 (1974)MathSciNetGoogle Scholar
  20. 20.
    Johnson, E.L.: Characterization of facets for multiple right-hand side choice linear programs. Math. Program. Study 14, 137–179 (1981)Google Scholar
  21. 21.
    Kılınç-Karzan, F.: On minimal inequalities for mixed integer conic programs. Math. Oper. Res. (2015). doi:10.1287/moor.2015.0737 MATHGoogle Scholar
  22. 22.
    Kılınç-Karzan, F., Yang, B.: Sufficient conditions and necessary conditions for the sufficiency of cut-generating functions. Technical report, December (2015). http://www.andrew.cmu.edu/user/fkilinc/files/draft-sufficiency-web.pdf
  23. 23.
    Kılınç-Karzan, F., Yıldız, S.: Two-term disjunctions on the second-order cone. In: Lee, Jon, Vygen, Jens (eds.) IPCO, Volume 8494 of Lecture Notes in Computer Science, pp. 345–356. Springer, Heidelberg (2014)Google Scholar
  24. 24.
    Kılınç-Karzan, F., Yıldız, S.: Two term disjunctions on the second-order cone. Math. Program. 154, 463–491 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Modaresi, S., Kılınç, M.R., Vielma, J.P.: Intersection cuts for nonlinear integer programming: convexification techniques for structured sets. Math. Program. (2015). doi:10.1007/s10107-015-0866-5 MATHGoogle Scholar
  26. 26.
    Modaresi, S., Kılınç, M.R., Vielma, J.P.: Split cuts and extended formulations for mixed integer conic quadratic programming. Oper. Res. Lett. 43(1), 10–15 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Morán R, D.A., Dey, S.S., Vielma, J.P.: A strong dual for conic mixed-integer programs. SIAM J. Optim. 22(3), 1136–1150 (2012)Google Scholar
  28. 28.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)CrossRefMATHGoogle Scholar
  29. 29.
    Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, New Jersey (1970)Google Scholar
  30. 30.
    Yıldız, S., Cornuéjols, G.: Disjunctive cuts for cross-sections of the second-order cone. Oper. Res. Lett. 43(4), 432–437 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Mathematics and StatisticsOakland UniversityRochesterUSA

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