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Integer programming duality: Price functions and sensitivity analysis

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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.

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References

  1. R.E. Alcaly and A.V. Klevorick, “A note on dual prices of integer programs”,Econometrica 34 (1966) 206–214.

    Google Scholar 

  2. A. Bachem and R. Schrader, “A note on a theorem of Jeroslow”, Report No. 7897, Institut für Operations Research, Bonn.

  3. W.J. Baumol and R.E. Gomory, “Integer programming and pricing”,Econometrica 28 (1960) 521–550.

    Google Scholar 

  4. D.E. Bell and J.F. Shapiro, “A convergent duality theory for integer programming”,Operations Research 25 (1977) 419–434.

    Google Scholar 

  5. J.F. Benders, “Partitioning procedures for solving mixed variables programming problems”,Numerische Mathematik 4 (1962) 238–252.

    Google Scholar 

  6. C.E. Blair, “Extensions of subadditive functions used in cutting plane theory”, MSRR No. 360, Carnegie Mellon University (December 1974).

  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. 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. 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. V. Chvatal, “Edmonds polytopes and a hierarchy of combinatorial problems”,Discrete Mathematics 4 (1973) 305–337.

    Google Scholar 

  11. R.J. Dakin, “A tree-search algorithm for mixed integer programming”,Computer Journal 8 (1965) 250–255.

    Google Scholar 

  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. 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. J. Edmonds and W. Pulleyblank,Optimum matching theory (Johns Hopkins Press, to appear).

  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. A.M. Geoffrion, “Lagrangean relaxation for integer programming”,Mathematical Programming Study 2 (1974) 82–114.

    Google Scholar 

  17. A.M. Geoffrion and R. Nauss, “Parametric and postoptimality analysis in integer linear programming”,Management Science 23 (1977) 453–466.

    Google Scholar 

  18. R.E. Gomory, “An algorithm for the mixed integer problem”, RM-2597, RAND Corp. (1960).

  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. R.E. Gomory, “Some polyhedra related to combinatorial problems”,Linear Algebra and Its Applications 2 (1969) 451–558.

    Google Scholar 

  21. R.E. Gomory and E.L. Johnson, “Some continuous functions related to corner polyhedron”,Mathematical Programming 3 (1972) 23–85.

    Google Scholar 

  22. R.G. Jeroslow, “Minimal inequalities”,Mathematical Programming 17 (1979) 1–15.

    Google Scholar 

  23. R.G. Jeroslow, “Cutting plane theory: algebraic methods”,Discrete Mathematics 23 (1978) 121–150.

    Google Scholar 

  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. 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. E.L. Johnson, “On the group problem for mixed integer programming”,Mathematical Programming Study 2 (1974) 137–179.

    Google Scholar 

  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. 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. T.C. Koopmans, “Concepts of optimality and their uses”, Nobel Memorial Lecture, 11 December 1975,Mathematical Programming 11 (1976) 212–228.

    Google Scholar 

  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. 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).

  32. H.P. Williams, “The economic interpretation of duality for practical mixed integer programming problems”, Mimeo University of Edinburgh (June 1977).

  33. L.A. Wolsey, “Integer programming duality: A view of some recent developments”, Mimeo, CORE (Louvain-la-Neuve, August 1978).

    Google Scholar 

  34. L.A. Wolsey, “Decomposition algorithms for general mathematical programs”, CORE Discussion Paper 7940 (Louvain-la-Neuve, December 1979).

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Authors and Affiliations

  1. C.O.R.E., Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Laurence A. Wolsey

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  1. Laurence A. Wolsey
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Additional information

This research was supported in part by a Senior Visiting Research Fellowship from the Science Research Council at the London School of Economics while the author was on leave from CORE, Université Catholique de Louvain, at Louvain-la-Neuve.

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Wolsey, L.A. Integer programming duality: Price functions and sensitivity analysis. Mathematical Programming 20, 173–195 (1981). https://doi.org/10.1007/BF01589344

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  • DOI: https://doi.org/10.1007/BF01589344

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