Summary
Following an introduction, which discusses the motivation for studying integer programming, the relevance of computational complexity and the relative merits of integer and dynamic programming, the branch and bound method is introduced in general terms. Various types of global entity to which it can be applied are introduced. These are integer variables, semicontinuous variables, special ordered sets and chains of linked ordered sets. A discussion of the algorithmic details follows. Finally, various approaches to automatic model reformulation are discussed: this seems to be the most important current area of integer programming research.
Keywords
- Linear Programming Problem
- Lagrangian Relaxation
- Integer Variable
- Integer Programming Problem
- Shadow Prex
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
E.M.L. Beale (1968) Mathematical Programming in Practice (Pitmans. London).
E.M.L. Beale (1980) “Branch and bound methods for numerical optimization of non-convex functions” COMPSTAT 80 Proceedings in Computational Statistics Edited by M.M. Barritt and D. Wishart pp 11–20 (Physica Verlag. Wien).
E.M.L. Beale (1983) “A mathematical programming model for the long-term development of an off-shore gas field” Discrete Applied Mathematics 5 pp 1–9.
E.M.L. Beale and J.J.H. Forrest (1976) “Global optimization using special ordered sets” Mathematical Programming 10 pp 52–69.
E.M.L. Beale and J.A. Tomlin (1970) “Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables” in Proceedings of the Fifth International Conference on Operational Research Ed. J. Lawrence pp 447–454. (Tavistock Publications. London).
A.L. Brearley, G. Mitra and H.P. Williams (1975) “Analysis of mathematical programming problems prior to applying the simplex method.” Mathematical Programming 8 pp 54–83.
H. Crowder, E.L. Johnson and M.W. Padberg (1983) “Solving large-scale zero-one linear programming problems” Operations Research 31 pp 803–834.
N. Driebeek (1966) “An algorithm for the solution of mixed integer programming problems” Management Science 12 pp 576–587.
J.J.H. Forrest, J.P.H. Hirst and J.A. Tomlin (1974) “Practical solution of large mixed integer programming problems with UMPIRE” Management Science 20 pp 736–773.
A.M. Geoffrion and R.E. Marsten (1972) “Integer Programming Algorithms: A framework and state-of-the-art-survey” Management Science 18 pp 465–491.
R.E. Gomory (1958) “Outline of an algorithm for integer solutions to linear programs” Bulletin of the American Mathematical Society 64 pp-275–278.
A.H. Land and S. Powell (1979) “Computer codes for problems of integer programming” in Annals of Discrete Mathematics 5; Discrete Optimization Edited by P.L. Hammer, E.L. Johnson and B.H. Korte. pp 221–269 (North Holland Publishing Company. Amsterdam).
K.L. Hoffman and M.W. Padberg (1985) “LP-Based combinatorial problem solving”. This volume.
J.D.C. Little, K.C. Murty, D.W. Sweeney and C. Karel (1963) “An algorithm for the traveling salesman problem” Operations Research 11 pp 972–989.
H.M. Markowitz and A.S. Manne (1957) “On the solution of discrete programming problems” Econometrica 25 pp 84–110.
C.E. Miller (196 3) “The simplex method for local separable programming” in Recent Advances in Mathematical Programming Edited by R.L. Graves and P. Wolfe pp 89–100 (McGraw Hill. New York).
A.H.G. Rinnooy and G.T. Timmer (1985) “Stochastic methods for global optimization” This volume.
J.A. Tomlin (19 71) “An improved branch and bound method for integer programming” Operations Research 19 pp 1070–1075.
H.P. Williams (1978) Model Building in Mathematical Programming (Wiley. Chichester).
H.P. Williams (1985) “Model Building in Linear and Integer Programming” This volume.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beale, E.M.L. (1985). Integer Programming. In: Schittkowski, K. (eds) Computational Mathematical Programming. NATO ASI Series, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82450-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-82450-0_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-82452-4
Online ISBN: 978-3-642-82450-0
eBook Packages: Springer Book Archive