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.
Notes
Note that when \(E=\mathbb {R}^n\), and a linear map \(A:\mathbb {R}^n\rightarrow \mathbb {R}^m\) is just an \(m\times n\) real-valued matrix, and its conjugate is given by its transpose, i.e., \(A^*=A^T\).
We note that our definition of tightness of an inequality does not require the corresponding hyperplane to have a nonempty intersection with the feasible region, as is sometimes the definition used in the literature.
References
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)
Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Math. Program. 122(1), 1–20 (2010)
Bachem, A., Johnson, E.L., Schrader, R.: A characterization of minimal valid inequalities for mixed integer programs. Oper. Res. Lett. 1, 63–66 (1982)
Bachem, A., Schrader, R.: Minimal inequalities and subadditive duality. SIAM J. Control Optim. 18, 437–443 (1980)
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)
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)
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)
Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization. SIAM, Philadehia (2001)
Bienstock, D., Michalka, A.: Cutting-planes for optimization of convex functions over nonconvex sets. SIAM J. Optim. 24(2), 643–677 (2014)
Blair, C.E.: Minimal inequalities for mixed integer programs. Discrete Math. 24, 147–151 (1978)
Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34(3), 538–546 (2009)
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
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)
Cornuéjols, G., Wolsey, L., Yıldız, S.: Sufficiency of cut-generating functions. Math. Program. 152, 643–651 (2015)
Dadush, D., Dey, S.S., Vielma, J.P.: The split closure of a strictly convex body. Oper. Res. Lett. 39, 121–126 (2011)
Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra. Math. Program. 3, 23–85 (1972)
Jeroslow, R.G.: Cutting plane theory: algebraic methods. Discrete Math. 23, 121–150 (1978)
Jeroslow, R.G.: Minimal inequalities. Math. Program. 17, 1–15 (1979)
Johnson, E.L.: On the group problem for mixed integer programming. Math. Program. 2, 137–179 (1974)
Johnson, E.L.: Characterization of facets for multiple right-hand side choice linear programs. Math. Program. Study 14, 137–179 (1981)
Kılınç-Karzan, F.: On minimal inequalities for mixed integer conic programs. Math. Oper. Res. (2015). doi:10.1287/moor.2015.0737
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
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)
Kılınç-Karzan, F., Yıldız, S.: Two term disjunctions on the second-order cone. Math. Program. 154, 463–491 (2015)
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
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)
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)
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)
Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, New Jersey (1970)
Yıldız, S., Cornuéjols, G.: Disjunctive cuts for cross-sections of the second-order cone. Oper. Res. Lett. 43(4), 432–437 (2015)
Acknowledgments
The authors wish to thank the review team for their constructive feedback that improved the presentation of the material in this paper. The research of the first author is supported in part by NSF Grant CMMI 1454548.
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Kılınç-Karzan, F., Steffy, D.E. On sublinear inequalities for mixed integer conic programs. Math. Program. 159, 585–605 (2016). https://doi.org/10.1007/s10107-015-0968-0
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DOI: https://doi.org/10.1007/s10107-015-0968-0