Projection and Lifting in Combinatorial Optimization

  • Egon Balas
Chapter
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2241)

Abstract

This is an overview of the significance and main uses of projection and lifting in integer programming and combinatorial optimization. Its first four sections deal with those basic properties of projection that make it such an effiective and useful bridge between problem formulations in different spaces, i.e. different sets of variables. They discuss topics like the integrality-preserving property of projection, the dimension of projected polyhedra, when do facets of a polyhedron project into facets of its projection, and so on. They also describe the use of projection for comparing the strength of different formulations of the same problem, and for proving the integrality of polyhedra by using extended formulations. The next section of the survey deals with disjunctive programming, or optimization over unions of polyhedra, whose most important incarnation are mixed 0-1 programs and their partial relaxations. It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as well as the sequential convexification of facial disjunctive programs, among them mixed 0-1 programs, with the related concept of disjunctive rank. Finally, the last two sections review the recent developments in disjunctive programming, namely lift-and-project cuts, the construction of cut generating linear programs, techniques for lifting and for strengthening disjunctive cuts, and the embedding of these cuts into a branch-and-cut framework, along with a brief outline of computational results.

Keywords

Convex Hull Valid Inequality Incidence Vector Projection Cone Discrete Apply Mathematic 
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 to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Balas, “Disjunctive Programming: Properties of the Convex Hull of Feasible Points.” Invited paper, with a foreword by G. Cornuéjols and W. Pulleyblank, Discrete Applied Mathematics, 89, 1998, 3–44 (originally MSRR# 348, Carnegie Mellon University, July 1974).Google Scholar
  2. 2.
    E. Balas, “Disjunctive Programming.” Annals of Discrete Mathematics, 5, 1979, 3–51.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    E. Balas, “Disjunctive Programming and a Hierarchy of Relaxations for Discrete Optimization Problems.” SIAM Journal on Algebraic and Discrete Methods, 6, 1985, 466–485.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    E. Balas, “The Assignable Subgraph Polytope of a Directed Graph.” Congressus Numerantium, 60, 1987, 35–42.MathSciNetGoogle Scholar
  5. 5.
    E. Balas, “A Modified Lift-and-Project Procedure.” Mathematical Programming, 79, 1997, 19–31.MathSciNetGoogle Scholar
  6. 6.
    E. Balas, “Projection with a Minimal System of Inequalities.” Computational Optimization and Applications, 10, 1998, 189–193.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    E. Balas, S. Ceria and G. Cornuéjols, “A Lift-and-Project Cutting Plane Algorithm for Mixed 0-1 Programs.” Mathematical Programming, 58, 1993, 295–324.CrossRefMathSciNetGoogle Scholar
  8. 8.
    E. Balas, S. Ceria and G. Cornuéjols, “Mixed 0-1 Programming by Lift-and-Project in a Branch-and-Cut Framework.” Management Science, 42, 1996, 1229–1246.MATHGoogle Scholar
  9. 9.
    E. Balas, S. Ceria, G. Cornuéjols and G. Pataki, “Polyhedral Methods for the Maximum Clique Problem.” In D. Johnson and M. Trick (editors), Clique, Coloring and Satisfiability: The Second DIMACS Challenge. The American Mathematical Society, Providence, RI, 1996, 11–27.Google Scholar
  10. 10.
    E. Balas and R.G. Jeroslow, “Strengthening Cuts for Mixed Integer Programs.” European Journal of Operational Research, 4, 1980, 224–234.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    E. Balas and J.B. Mazzola, “Nonlinear 0-1 Programming: I. Linearization Techniques, II. Dominance Relations and Algorithms.” Mathematical Programming, 30, 1984, 1–21 and 22-45.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    E. Balas and M. Oosten, “On the Dimension of Projected Polyhedra.” Discrete Applied Mathematics, 87, 1998, 1–9.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    E. Balas and M. Oosten, “The Cycle Polytope of a Directed Graph.” Networks, 36, 2000, 34–46.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    E. Balas and W.R. Pulleyblank, “The Perfectly Matchable Subgraph Polytope of a Bipartite Graph.” Networks, 13, 1983, 495–518.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    E. Balas and W.R. Pulleyblank, “The Perfectly Matchable Subgraph Polytope of an Arbitrary Graph.” Combinatorica, 9, 1989, 321–337.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    E. Balas, J. Tama and J. Tind, “Sequential Convexification in Reverse Convex and Disjunctive Programming.” Mathematical Programming, 44, 1989, 337–350.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    N. Beaumont, “An Algorithm for Disjunctive Programming.” European Journal of Operational Research, 48, 1990, 362–371.MATHCrossRefGoogle Scholar
  18. 18.
    R. Bixby, W. Cook, A. Cox and E. Lee, “Computational Experience with Parallel Mixed Integer Programming in a Distributed Environment.” Annals of Operations Research, 90, 1999, 19–45.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    C. Blair, “Two Rules for Deducing Valid Inequalities for 0-1 Programs.” SIAM Journal of Applied Mathematics, 31, 1976, 614–617.CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    C. Blair, “Facial Disjunctive Programs and Sequences of Cutting Planes.” Discrete Applied Mathematics, 2, 1980, 173–180.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    S. Ceria and G. Pataki, “Solving Integer and Disjunctive Programs by Lift-and-Project.” In R.E. Bixby, E.A. Boyd and R.Z. Rios-Mercado (editors), IPCO VI, Lecture Notes in Computer Science, 1412, Springer 1998, 271–283.Google Scholar
  22. 22.
    S. Ceria and J. Soares, “Disjunctive Cuts for Mixed 0-1 Programming: Duality and Lifting.” GSB, Columbia University, 1997.Google Scholar
  23. 23.
    S. Ceria and J. Soares, “Convex Programming for Disjunctive Convex Optimization.” Mathematical Programming, 86, 1999, 595–614.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    G.B. Dantzig, R.D. Fulkerson and S.M. Johnson, “Solution of a Large-Scale Traveling Salesman Problem.” Operations Research, 2, 1954, 393–410.MathSciNetCrossRefGoogle Scholar
  25. 25.
    R. Fortet, “Applications de l’algébre de Boolean recherche opérationnelle.” Revue Francaise de Recherche Opérationnelle, 4, 1960, 17–25.Google Scholar
  26. 26.
    R.G. Jeroslow, “Cutting Plane Theory: Disjunctive Methods.” Annals of Discrete Mathematics, 1, 1977, 293–330.MathSciNetCrossRefGoogle Scholar
  27. 27.
    R.G. Jeroslow, “Representability in Mixed Integer Programming I: Characterization Results.” Discrete Applied Mathematics, 17, 1987, 223–243.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    R.G. Jeroslow, “Logic Based Decision Support: Mixed Integer Model Formulation.” Annals of Discrete Mathematics, 40, 1989.Google Scholar
  29. 29.
    E.L. Johnson, “The Group Problem for Mixed-Integer Programming.” Mathematical Programming Study, 2, 1974, 137–179.Google Scholar
  30. 30.
    A. Letchford, “On Disjunctive Cuts for Combinatorial Optimization.” Journal of Combinatorial Optimization, 5, 2001 (Issue 3).Google Scholar
  31. 31.
    L. Lovász and A. Schrijver, “Cones of Matrices and Set Functions and 0-1 Optimization.” SIAM Journal of Optimization, 1, 1991, 166–190.MATHCrossRefGoogle Scholar
  32. 32.
    C.E. Miller, A.W. Tucker and R.A. Zemlin, “Integer Programming Formulations and Traveling Salesman Problems.” Journal of the ACM, 7, 1960, 326–329.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    G. Nemhauser and L. Wolsey, Integer and Combinatorial Optimization, Wiley, 1986.Google Scholar
  34. 34.
    M. Padberg and Ting-Yi Sung, “An Analytical Comparison of Difierent Formulations of the Traveling Salesman Problem.” Math. Programming, 52, 1991, 315–357.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    G. Pataki, personal communication, March 2001.Google Scholar
  36. 36.
    M. Perregaard and E. Balas, “Generating Cuts from Multiple-Term Disjunctions.” In K. Aardal and B. Gerards (Editors), Integer Programming and Combinatorial Optimization (8th IPCO Conference, Utrecht, 2001), LCNS No. 2081, Springer, 2001.Google Scholar
  37. 37.
    H. Sherali and W. Adams, “A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems.” SIAM Journal on Discrete Mathematics, 3, 1990, 411–430.MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    H. Sherali and C. Shetty, Optimization with Disjunctive Constraints. Lecture Notes in Economics and Mathematical Systems, 181, Springer, 1980.Google Scholar
  39. 39.
    J. Stoer and C. Witzgall, Convexity and Optimization in Finite Dimensions I. Springer, 1970.Google Scholar
  40. 40.
    R.A. Stubbs and S. Mehrotra, “A Branch-and-Cut Method for 0-1 Mixed Convex Programming.” Department of Industrial Engineering, Northwestern University, 1996.Google Scholar
  41. 41.
    S. Thienel, “ABACUS: A Branch-and-Cut System.” Doctoral Dissertation, Faculty of Mathematics and The Natural Sciences, University of Cologne, 1995.Google Scholar
  42. 42.
    M. Turkay and I.E. Grossmann, “Disjunctive Programming Techniques for the Optimization of Process Systems with Discontinuous Investment Costs-Multiple Size Regions.” Industrial Engineering Chemical Research, 35, 1996, 2611–2623.CrossRefGoogle Scholar
  43. 43.
    H.P. Williams, “An Alternative Explanation of Disjunctive Formulations.” European Journal of Operational Research, 72, 1994, 200–203.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Egon Balas
    • 1
  1. 1.Mathematical Sciences DepartmentCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations