Abstract
We provide a characterization of minimal inequalities for bounded mixed integer programs, in terms of subadditive functions. The condition on the columns of the integer-constrained variables is analogous to that obtained earlier for the group problem, and we also determine the condition on the columns of the continuous variables.
Similar content being viewed by others
References
A.A. Aroaz (Durand), “Polyhedral neopolarities”, Ph.D. dissertation, University of Waterloo (Waterloo, Canada, November 1973).
A.A. Aroaz (Durand) and J. Edmonds, “The faces of master covering polyhedra”, page B-351 of the ORSA/TIMS Bulletin for a meeting in Atlantic City (November 1972).
E. Balas, “Intersection cuts from disjunctive constraints”, Management Science Res. Rep. no. 330, GSIA, Carnegie-Mellon University (first draft August 1973, revised and expanded February 1974). A shortened version of much of this report appeared in [4].
E. Balas, “Disjunctive programming: cutting planes from logical conditions”, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear programming 2 (Academic Press, New York, 1975) pp. 279–312.
E. Balas, “Disjunctive programming: Properties of the convex hull of feasible points”, Management Science Res. Rep. no. 348, GSIA, Carnegie-Mellon University (July 1974). Some of this material will appear in [6].
E. Balas, “Disjunctive programming”,Annals of Discrete Mathematics 5 (1979) 3–52.
C.E. Blair, “Minimal inequalities for mixed integer programs”,Discrete Mathematics 24 (1978) 147–151.
C.E. Blair and R.G. Jeroslow, “The value function of a mixed integer program: I”,Discrete Mathematics 19 (1977) 121–138.
C.E. Blair and R.G. Jeroslow, “The value function of a mixed integer program: II”,Discrete Mathematics 25 (1979) 7–19.
C.-A. Burdet and E.L. Johnson, “A subadditive approach to solve linear integer programs”, IBM Research RC5507 (Yorktown Heights, NY, July 1975).
F. Glover, “Polyhedral annexation in mixed integer and combinatorial programming”,Mathematical Programming 9 (1975) 161–188.
R.E. Gomory, “On the relation between integer and non-integer solutions to linear programs”,Proceedings of the National Academy of Sciences 53 (1965) 260–265.
R.E. Gomory, “Some polyhedra related to combinatorial problems”,Linear Algebra and Its Applications 2 (1969) 541–558.
R.E. Gomory and E.L. Johnson, “Some continuous functions related to corner polyhedra, I and II”,Mathematical Programming 3 (1972) 23–85 and 359–389.
R.G. Jeroslow, “The principles of cutting-plane theory: Part I”, with an Addendum, GSIA, Carnegie-Mellon University (February 1974). Also issued as Management Science Res. Rep. no. 332 (March 1974).
R.G. Jeroslow, “Cutting-plane theory: algebraic methods”,Discrete Mathematics 23 (1978) 121–150.
R.G. Jeroslow, “Minimal inequalities”, Management Science Research Report no. 362, GSIA, Carnegie-Mellon University (March 1975, revised April 1975).
E.L. Johnson, “The group problem for mixed integer programming”,Mathematical Programming Study 2 (December 1974) 137–179.
E.L. Johnson, “Integer programs with continuous variables” (July 1974).
E.L. Johnson, “Faces of polyhedra of mixed integer programming problems”,Istituto Nazionale di alta Matematica XIX (1976) 289–299.
E.L. Johnson, “Cyclic groups, cutting planes, shortest paths”, in: T.C. Hu and S.M. Robinson, eds.,Mathematical programming (Academic Press, New York, 1973) pp. 185–211.
Author information
Authors and Affiliations
Additional information
The research reported here was done at Carnegie-Mellon University.
The report was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University, under Contract N00014-67-A-0314-0007 NR 047-048 with U.S. Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.
Rights and permissions
About this article
Cite this article
Jeroslow, R.G. Minimal inequalities. Mathematical Programming 17, 1–15 (1979). https://doi.org/10.1007/BF01588222
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01588222