Abstract
Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Günlük (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.
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References
Aczél J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966)
Aráoz J., Evans L., Gomory R.E., Johnson E.L.: Cyclic groups and knapsack facets. Math. Program. 96, 377–408 (2003)
Dash S., Günlük O.: Valid inequalities based on simple mixed-integer sets. In: Nemhauser, G.L., Bienstock, D.(eds) Proceedings 10th Conference on Integer Programming and Combinatorial Optimization, pp. 33–45. Springer, Heidelberg (2004)
Dash S., Günlük O.: Valid inequalities based on simple mixed integer set. Math. Program. 106, 29–53 (2006)
Dash S., Günlük O.: Valid inequalities based on the interpolation procedure. Math. Program. 106, 111–136 (2006)
Dey, S.S.: Strong cutting planes for unstructured mixed integer programs using multiple constraints. PhD Thesis, Purdue University, West Lafayette (2007)
Dey S.S., Richard J.-P.P.: Sequential-merge facets for two-dimensional group problems. In: Fischetti, M., Williamson, D.P.(eds) Proceedings 12th Conference on Integer Programming and Combinatorial Optimization, pp. 30–42. Springer, Heidelberg (2007)
Dey S.S., Richard J.-P.P.: Facets of two-dimensional infinite group problems. Math. OR 33, 140–166 (2008)
Gomory R.E.: Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2, 341–375 (1969)
Gomory R.E., Johnson E.L.: Some continuous functions related to corner polyhedra, part I. Math. Program. 3, 23–85 (1972)
Gomory R.E., Johnson E.L.: Some continuous functions related to corner polyhedra, part II. Math. Program. 3, 359–389 (1972)
Gomory R.E., Johnson E.L.: T-space and cutting planes. Math. Program. 96, 341–375 (2003)
Gomory R.E., Johnson E.L., Evans L.: Corner polyhedra and their connection with cutting planes. Math. Program. 96, 321–339 (2003)
Johnson E.L.: On the group problem for mixed integer programming. Math. Program. Study 2, 137–179 (1974)
Letchford A.N., Lodi A.: Strengthening Chvátal–Gomory cuts and gomory fractional cuts. Oper. Res. Lett. 30(2), 74–82 (2002)
Miller L.A., Li Y., Richard J.-P.P.: New facets for finite and infinite group problems from approximate lifting. Naval Res. Logis. 55, 172–191 (2008)
Richard, J.-P.P., Li, Y., Miller, L.A.: Valid inequalities for MIPs and group polyhedra from approximate liftings. Math. Program. (2008) (to appear)
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S.S. Dey and J.-P.P. Richard was supported by NSF Grant DMI-03-48611.
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Dey, S.S., Richard, JP.P., Li, Y. et al. On the extreme inequalities of infinite group problems. Math. Program. 121, 145–170 (2010). https://doi.org/10.1007/s10107-008-0229-6
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DOI: https://doi.org/10.1007/s10107-008-0229-6