Generalized Chvátal-Gomory closures for integer programs with bounds on variables
Tóm tắt
Integer programming problems that arise in practice often involve decision variables with one or two sided bounds. In this paper, we consider a generalization of Chvátal-Gomory inequalities obtained by strengthening Chvátal-Gomory inequalities using the bounds on the variables. We prove that the closure of a rational polyhedron obtained after applying the generalized Chvátal-Gomory inequalities is also a rational polyhedron. This generalizes a result of Dunkel and Schulz on 0–1 problems to the case when some of the variables have upper or lower bounds or both while the rest of them are unbounded.
Tài liệu tham khảo
Bockmayr, A., Eisenbrand, F.: Cutting planes and the elementary closure in fixed dimension. Math. Oper. Res. 26, 304–312 (2001)
Bonami, P., Lodi, A., Tramontani, A., Wiese, S.: Cutting planes from wide split disjunction, IPCO 2017. In: Eisenbrand, F., Könemann, J. (eds.) LNCS 10328, pp. 99–110 (2017)
Crowder, H., Johnson, E., Padberg, M.: Solving large-scale zero-one linear programming problems. Oper. Res. 31, 803–834 (1983)
Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discret. Math. 4, 305–337 (1973)
Cook, W.J., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)
Dadush, D., Dey, S.S., Vielma, J.P.: On the Chvátal-Gomory closure of a compact convex set. Math. Program. 145, 327–348 (2014)
Dash, S., Günlük, O., Moran, D.A.R.: On the polyhedrality of closures of multi-branch split sets and other polyhedra with bounded max-facet-width. SIAM J. Optim. 27, 1340–1361 (2017)
Dash, S., Günlük, O., Moran, D.A.R.: Lattice closures of polyhedra. In: Mathematical Programming
Del Pia, A., Gijswijt, D., Linderoth, J., Zhu, H.: Integer packing sets form a well-quasi-ordering. arXiv:1911.12841 (2019)
Dirichlet, G.L.: Verallgemeinerung eines Satzes aus der Lehre von den Kettenbriichen nebst einigen Anwendungen auf die Theorie der Zahlen. Bericht iiber die zur Bekanntmachung geeigneten Verhandlungen der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1842) 93–95. (Reprinted in: L. Kronecker (ed.) G.L. Dirichlet’s Werke Vol. I, G. Reimer, Berlin, 1889 (reprinted: Chelsea, New York, 1969), pp. 635–638)
Dunkel, J., Schulz, A.S.: A refined Gomory-Chvátal closure for polytopes in the unit cube. Technical report. http://www.optimization-online.org/DB_HTML/2012/03/3404.html (2012). Accessed 7 June 2020
Dunkel, J., Schulz, A.S.: The Gomory-Chvátal closure of a nonrational polytope is a rational polytope. Math. Oper. Res. 38, 63–91 (2013)
Eisenbrand, F.: On the membership problem for the elementary closure of a polyhedron. Combinatorica 19, 297–300 (1999)
Fischetti, M., Lodi, A.: On the knapsack closure of 0–1 integer linear programs. Electron. NotesDiscret. Math. 36, 799–804 (2010)
Furini, F., Ljubić, I., Sinnl, M.: ILP and CP formulations for the lazy bureaucrat problem, CPAIOR 2015. In: Michel, L. (ed.) LNCS, vol. 9075, pp. 255–270 (2015)
Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64, 275–278 (1958)
Huchette, J.: Advanced mixed-integer programming formulations: methodology, computation, and application, Ph.D. thesis, Massachusetts Institute of Technology (2018)
Meyer, R.R.: On the existence of optimal solutions to integer and mixed integer programming problems. Math. Program. 7, 223–235 (1974)
Pashkovich, K., Poirrier, L., Pulyassary, H.: The aggregation closure is polyhedral for packing and covering integer programs arXiv:1910.03404 (2019)
Pokutta, S.: Lower bounds for Chvátal-Gomory style operators. Technical report. http://www.optimization-online.org/DB_HTML/2011/09/3151.html (2011). Accessed 7 June 2020
Schrijver, A.: On cutting planes. Ann. Discret. Math. 9, 291–296 (1980)
Vielma, J.P.: Embedding formulations and complexity for unions of polyhedra. Manag. Sci. 64, 4471–4965 (2018)