Annals of Operations Research

, Volume 139, Issue 1, pp 95–129 | Cite as

Depth-Optimized Convexity Cuts

Article

Abstract

This paper presents a general, self-contained treatment of convexity or intersection cuts. It describes two equivalent ways of generating a cut—via a convex set or a concave function—and a partial-order notion of cut strength. We then characterize the structure of the sets and functions that generate cuts that are strongest with respect to the partial order. Next, we specialize this analytical framework to the case of mixed-integer linear programming (MIP). For this case, we formulate two kinds of the deepest cut generation problem, via sets or via functions, and subsequently consider some special cases which are amenable to efficient computation. We conclude with computational tests of one of these procedures on a large set of MIPLIB problems.

Keywords

integer programming cutting planes convexity cuts 

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References

  1. Balas, E. (1971). “Intersection Cuts—A New Type of Cutting Planes for Integer Programming.” Operations Research 19, 19–39.Google Scholar
  2. Balas, E. (1972). “Integer Programming and Convex Analysis: Intersection Cuts from Outer Polars.” Mathematical Programming 2, 330–382.CrossRefGoogle Scholar
  3. Balas, E., V.J. Bowman, F. Glover, and D. Sommer. (1971). “An Intersection Cut From the Dual of the Unit Hypercube.” Operations Research 19, 40–44.Google Scholar
  4. Balas, E., S. Ceria, and G. Cornuéjols. (1993). “A Lift-And-Project Cutting Plane Algorithm for Mixed 0–1 Programs.” Mathematical Programming 58(3), 295–324.CrossRefGoogle Scholar
  5. Balas, E., S. Ceria, and G. Cornuéjols. (1996). “Mixed 0–1 Programming by Lift-And-Project in a Branch-And-Cut Framework.” Management Science 42(9), 1229–1246.Google Scholar
  6. Balas, E., S. Ceria, G. Cornuéjols, and N. Natraj. (1996). “Gomory Cuts Revisited.” Operations Research Letters 19(1), 1–9.CrossRefGoogle Scholar
  7. Balas, E. and R.G. Jeroslow. (1980). “Strengthening Cuts for Mixed Integer Programs.” European Journal of Operational Research, 4(4), 224–234.CrossRefGoogle Scholar
  8. Balas, E. and M. Perregaard. (2003). “A Precise Correspondence Between Lift-And-Project Cuts, Simple Disjunctive Cuts, and Mixed Integer Gomory Cuts for 0–1 Programming.” Mathematical Programming 94(2–3, Ser. B), 221–245.CrossRefGoogle Scholar
  9. Bixby, R.E., S. Ceria, C.M. McZeal, and M.W.P. Savelsberg. (1998). “An Updated Mixed Integer Programming Library: MIPLIB 3.0.” Technical Report 98-3, Department of Computational and Applied Mathematics, Rice University.Google Scholar
  10. COmputational INfrastructure for Operations Research, 2004. http://www.coin-or.org/.
  11. Berg, M., M. Kreveld, M. Overmars, and O. Schwarzkopf. (2000). “Computational Geometry: Algorithms and Applications.” Springer-Verlag, Berlin, second, revised edition.Google Scholar
  12. Cornuéjols, G. and Y. Li. (2001). “Elementary Closures for Integer Programs.” Operations Research Letters 28(1), 1–8.CrossRefGoogle Scholar
  13. Glover, F. (1973). “Convexity Cuts and Cut Search.” Operations Research 21, 123–134.Google Scholar
  14. Jeroslow, R.G. (1977). “Cutting-Plane Theory: Disjunctive Methods.” In Studies in integer programming (Proc. Workshop, Bonn, 1975), Annals of Discrete Mathematics, North-Holland, Amsterdam, vol. 1, pp. 293–330.Google Scholar
  15. Lougee-Heimer, R. (2003). The Common Optimization Interface for Operations Research: Promoting Open-Source Software in the Operations Research Community. IBM Journal of Research and Development 47(1), 57–66. http://[0]//www.research.ibm.com/[0]journal/rd47-1.html.
  16. Nediak, M. and J. Eckstein. (2001). “Pivot, Cut, and Dive: A Heuristic for 0–1 Mixed Integer Programming.” RUTCOR Research Report 53-2001. Rutgers University, Piscataway, NJ.Google Scholar
  17. Owen, J.H. and S. Mehrotra. (2001). “A Disjunctive Cutting Plane Procedure for General Mixed-Integer Linear Programs.” Mathematical Programming 89(3), 437–448.CrossRefGoogle Scholar
  18. Raghavachari, M. (1969). “On Connections Between Zero-One Integer Programming and Concave Programming Under Linear Constraints.” Operations Research 17, 680–684.Google Scholar
  19. Rockafellar, R.T. (1970). Convex analysis. Princeton University Press, Princeton, N.J.Google Scholar
  20. Sherali, H.D. and W.P. Adams. (1994). A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-Integer Zero-One Programming Problems. Discrete Applied Mathematics 52(1), 83–106.CrossRefGoogle Scholar
  21. Tuy, H. (1964). “Concave Programming Under Linear Constraints.” Soviet Mathematics pp. 1437–1440.Google Scholar
  22. Zwart, P.B. (1973). “Nonlinear Programming: Counterexamples to Two Global Optimization Algorithms.” Operations Research 21(6), 1260–1266.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Business School and RUTCORRutgers UniversityPiscatawayUSA
  2. 2.School of BusinessQueen's UniversityKingstonCanada

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