Advertisement

Mathematical Programming

, Volume 20, Issue 1, pp 173–195 | Cite as

Integer programming duality: Price functions and sensitivity analysis

  • Laurence A. Wolsey
Article

Abstract

Recently a duality theory for integer programming has been developed. Here we examine some of the economic implications of this theory, in particular the necessity of using price functions in place of prices, and the possibility of carrying out sensitivity analysis of optimal solutions. In addition we consider the form of price functions that are generated by known algorithms for integer programming.

Key words

Integer Programming Duality Sensitivity Analysis Valid Inequalities Branch and Bound Methods Group Problems Lagrangean Relaxation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R.E. Alcaly and A.V. Klevorick, “A note on dual prices of integer programs”,Econometrica 34 (1966) 206–214.Google Scholar
  2. [2]
    A. Bachem and R. Schrader, “A note on a theorem of Jeroslow”, Report No. 7897, Institut für Operations Research, Bonn.Google Scholar
  3. [3]
    W.J. Baumol and R.E. Gomory, “Integer programming and pricing”,Econometrica 28 (1960) 521–550.Google Scholar
  4. [4]
    D.E. Bell and J.F. Shapiro, “A convergent duality theory for integer programming”,Operations Research 25 (1977) 419–434.Google Scholar
  5. [5]
    J.F. Benders, “Partitioning procedures for solving mixed variables programming problems”,Numerische Mathematik 4 (1962) 238–252.Google Scholar
  6. [6]
    C.E. Blair, “Extensions of subadditive functions used in cutting plane theory”, MSRR No. 360, Carnegie Mellon University (December 1974).Google Scholar
  7. [7]
    C.E. Blair and R.G. Jeroslow, “The value function of a mixed integer program: I”,Discrete Mathematics 19 (1977) 121–138.Google Scholar
  8. [8]
    C.E. Blair and R.G. Jeroslow, “The value function of a mixed integer program: II”,Discrete Mathematics 25 (1979) 7–19.Google Scholar
  9. [9]
    C.A. Burdet and E.L. Johnson, “A subadditive approach to solve linear integer programs”,Annals of Discrete Mathematics 1 (1977) 117–144.Google Scholar
  10. [10]
    V. Chvatal, “Edmonds polytopes and a hierarchy of combinatorial problems”,Discrete Mathematics 4 (1973) 305–337.Google Scholar
  11. [11]
    R.J. Dakin, “A tree-search algorithm for mixed integer programming”,Computer Journal 8 (1965) 250–255.Google Scholar
  12. [12]
    J. Edmonds, “Maximum matching and a polyhedron with 0–1 vertices”,Journal Research National Bureau of Standards 69(B) (1965) 125–130.Google Scholar
  13. [13]
    J. Edmonds, “Some well-solved problems in combinatorial optimization”, in: B. Roy, ed.,Combinatorial programming: methods and applications (D. Reidel Publishing Co., Dordrecht, Holland, 1975).Google Scholar
  14. [14]
    J. Edmonds and W. Pulleyblank,Optimum matching theory (Johns Hopkins Press, to appear).Google Scholar
  15. [15]
    M.L. Fisher, W.D. Northrup and J.F. Shapiro, “Using duality to solve discrete optimization problems: theory and computational experience”,Mathematical Programming Study 3 (1975) 56–94.Google Scholar
  16. [16]
    A.M. Geoffrion, “Lagrangean relaxation for integer programming”,Mathematical Programming Study 2 (1974) 82–114.Google Scholar
  17. [17]
    A.M. Geoffrion and R. Nauss, “Parametric and postoptimality analysis in integer linear programming”,Management Science 23 (1977) 453–466.Google Scholar
  18. [18]
    R.E. Gomory, “An algorithm for the mixed integer problem”, RM-2597, RAND Corp. (1960).Google Scholar
  19. [19]
    R.E. Gomory, “An algorithm for integer solutions to linear programs”, in: R.L. Graves and P. Wolfe, eds.,Recent advances in mathematical programming (McGraw-Hill, New York, 1963).Google Scholar
  20. [20]
    R.E. Gomory, “Some polyhedra related to combinatorial problems”,Linear Algebra and Its Applications 2 (1969) 451–558.Google Scholar
  21. [21]
    R.E. Gomory and E.L. Johnson, “Some continuous functions related to corner polyhedron”,Mathematical Programming 3 (1972) 23–85.Google Scholar
  22. [22]
    R.G. Jeroslow, “Minimal inequalities”,Mathematical Programming 17 (1979) 1–15.Google Scholar
  23. [23]
    R.G. Jeroslow, “Cutting plane theory: algebraic methods”,Discrete Mathematics 23 (1978) 121–150.Google Scholar
  24. [24]
    D.S. Johnson, A. Demers, J.D. Ullman, M.R. Carey and R.L. Graham, “Worst case performance bounds for simple one dimensional packing algorithms”,SIAM Journal of Computing 3 (1974) 299–325.Google Scholar
  25. [25]
    E.L. Johnson, “Cyclic groups, cutting planes and shortest paths”, in: T.C. Hu and S. Robinson, eds.,Mathematical programming (Academic Press, New York, 1973).Google Scholar
  26. [26]
    E.L. Johnson, “On the group problem for mixed integer programming”,Mathematical Programming Study 2 (1974) 137–179.Google Scholar
  27. [27]
    E.L. Johnson, “On the group problem and a subadditive approach to integer programming”,Annals of Discrete Mathematics 5 (1979) 97–112.Google Scholar
  28. [28]
    R. Kannan and C.L. Monma, “On the complexity of integer programming problems”, Report No. 7780, Institut für Operations Research (Bonn, December 1977).Google Scholar
  29. [29]
    T.C. Koopmans, “Concepts of optimality and their uses”, Nobel Memorial Lecture, 11 December 1975,Mathematical Programming 11 (1976) 212–228.Google Scholar
  30. [30]
    R.R. Meyer, “On the existence of optimal solutions to integer and mixed integer programs”,Mathematical Programming 7 (1974) 223–235.Google Scholar
  31. [31]
    J. Tind and L.A. Wolsey, “A unifying framework for duality theory in mathematical programming”, CORE Discussion Paper 7834 (Louvain-la-Neuve, August 1978, revised November 1979).Google Scholar
  32. [32]
    H.P. Williams, “The economic interpretation of duality for practical mixed integer programming problems”, Mimeo University of Edinburgh (June 1977).Google Scholar
  33. [33]
    L.A. Wolsey, “Integer programming duality: A view of some recent developments”, Mimeo, CORE (Louvain-la-Neuve, August 1978).Google Scholar
  34. [34]
    L.A. Wolsey, “Decomposition algorithms for general mathematical programs”, CORE Discussion Paper 7940 (Louvain-la-Neuve, December 1979).Google Scholar

Copyright information

© North-Holland Publishing Company 1981

Authors and Affiliations

  • Laurence A. Wolsey
    • 1
  1. 1.C.O.R.E., Université Catholique de LouvainLouvain-la-NeuveBelgium

Personalised recommendations