Abstract
We study the facial structure of a polyhedron associated with the single node relaxation of network flow problems with additive variable upper bounds. This type of structure arises, for example, in network design/expansion problems and in production planning problems with setup times. We first derive two classes of valid inequalities for this polyhedron and give the conditions under which they are facet-defining. Then we generalize our results through sequence independent lifting of valid inequalities for lower-dimensional projections. Our computational experience with large network expansion problems indicates that these inequalities are very effective in improving the quality of the linear programming relaxations.
This research is supported, in part, by NSF Grant DMI-9700285 to the Georgia Institute of Technology.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
K. Aardal, Y. Pochet, and L. A. Wolsey. Capacitated facility location: Valid inequalities and facets. Mathematics of Operations Research, 20:562–582, 1995.
A. Atamtürk. Conflict graphs and flow models for mixed-integer linear optimization problems. PhD thesis, ISyE, Georgia Institute of Technology, Atlanta, USA, 1998.
A. Atamtürk. Flow packing facets of the single node fixed-charge flow polytope. Technical report, IEOR, University of California at Berkeley, 1998.
I. Barany, T. J. Van Roy, and L. A. Wolsey. Uncapacitated lot sizing: The convex hull of solutions. Mathematical Programming Study, 22:32–43, 1984.
D. Bienstock and O. Günlük. Capacitated network design-Polyhedral structure and computation. INFORMS Journal on Computing, 8:243–259, 1996.
Z. Gu, G. L. Nemhauser, and M. W. P. Savelsbergh. Lifted knapsack covers inequalities for 0–1 integer programs: Computation. Technical Report LEC-94-9, Georgia Institute of Technology, Atlanta GA, 1994. (to appear in INFORMS Journal on Computing).
Z. Gu, G. L. Nemhauser, and M. W. P. Savelsbergh. Sequence independent lifting. Technical Report LEC-95-08, Georgia Institute of Technology, Atlanta, 1995.
G. L. Nemhauser, M. W. P. Savelsbergh, and G. S. Sigismondi. MINTO, a Mixed INTeger Optimizer. Operations Research Letters, 15:47–58, 1994.
M. W. Padberg, T. J. Van Roy, and L. A. Wolsey. Valid linear inequalities for fixed charge problems. Operations Research, 32:842–861, 1984.
Y. Pochet. Valid inequalities and separation for capacitated economic lot sizing. Operations Research Letters, 7:109–115, 1988.
M. Stoer and G. Dahl. A polyhedral approach to multicommodity survivable network. Numerische Mathematik, 68:149–167, 1994.
T. J. Van Roy and L. A. Wolsey. Valid inequalities for mixed 0–1 programs. Discrete Applied Mathematics, 14:199–213, 1986.
L. A. Wolsey. Valid inequalities and superadditivity for 0/1 integer programs. Mathematics of Operations Research, 2:66–77, 1977.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Atamtürk, A., Nemhauser, G.L., Savelsbergh, M.W.P. (1999). Valid Inequalities for Problems with Additive Variable Upper Bounds. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1999. Lecture Notes in Computer Science, vol 1610. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48777-8_5
Download citation
DOI: https://doi.org/10.1007/3-540-48777-8_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66019-4
Online ISBN: 978-3-540-48777-7
eBook Packages: Springer Book Archive