Mathematical Programming

, Volume 20, Issue 1, pp 303–326 | Cite as

An advanced implementation of the Dantzig—Wolfe decomposition algorithm for linear programming

  • James K. Ho
  • Etienne Loute
Article
Received:
Revised:

Abstract

Since the original work of Dantzig and Wolfe in 1960, the idea of decomposition has persisted as an attractive approach to large-scale linear programming. However, empirical experience reported in the literature over the years has not been encouraging enough to stimulate practical application. Recent experiments indicate that much improvement is possible through advanced implementations and careful selection of computational strategies. This paper describes such an effort based on state-of-the-art, modular linear programming software (IBM's MPSX/370).

Key words

Large-Scale Systems Decomposition Algorithms Structured Linear Programs Optimization Software 
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]
    I. Alder and A. Ülkücü, “On the number of iterations in Dantzig—Wolfe decomposition”, in: D.M. Himmelblau, ed.,Decomposition of large scale problems (North-Holland, Amsterdam, 1973) pp. 181–187.Google Scholar
  2. [2]
    E. Beale, P. Huges and R. Small, “Experiences in using a decomposition program”,Computer Journal 8(1965) 13–18.Google Scholar
  3. [3]
    M. Benichou et al., “The efficient solution of large-scale linear programming problems-some algorithmic techniques and computational results”,Mathematical Programming 13 (1977) 280–332.Google Scholar
  4. [4]
    P. Broise, P. Huard and J. Sentenac,Decomposition des programmes mathématiques (Dunod, Paris, 1968).Google Scholar
  5. [5]
    CDC, “APEX-III Reference Manual”, Publication no. 76070000 (1974).Google Scholar
  6. [6]
    B. Culot, E. Loute and J.K. Ho, “DECOMPSX user's manual”, CORE computing report 80-B-01, C.O.R.E., Université Catholique de Louvain, Belgium, 1980. [In French.]Google Scholar
  7. [7]
    B. Culot and E. Loute, “DECOMPSX system manual”, CORE computing report 80-B-02, C.O.R.E., Université Catholique de Louvain, Belgium, 1980. [In French.].Google Scholar
  8. [8]
    B. Culot and E. Loute, “DECOMPSX source”, CORE computing report 80-B-03, C.O.R.E., Université Catholique de Louvain, Belgium, 1980.Google Scholar
  9. [9]
    G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, 1963).Google Scholar
  10. [10]
    G.B. Dantzig and P. Wolfe, “The decomposition principle for linear programs”,Operations Research 8 (1960) 101–111.Google Scholar
  11. [11]
    G.B. Dantzig and P. Wolfe, “The decomposition algorithm for linear programs”,Econometrica 29 (1961) 767–778.Google Scholar
  12. [12]
    G.B. Dantzig, “On the need for a system optimization laboratory”, in: R. Cottle and J. Krarup, eds.,Optimization methods for resource allocation (Crane—Russack, New York, 1974) pp. 3–24.Google Scholar
  13. [13]
    Y.M.I. Dirickx and L.P. Jennergren,Systems analysis by multilevel methods (Wiley, New York, 1979).Google Scholar
  14. [14]
    J. Forrest and J. Tomlin, “Updating triangular factors of the basis in the product form simplex method”,Mathematical Programming 2 (1972) 263–278.Google Scholar
  15. [15]
    R. Geottle, E. Cherniavsky and R. Tessmer, “An integrated multi-regional energy and interindustry model of the United states”, BNL 22728, Brookhaven National Laboratory, New York, May 1977.Google Scholar
  16. [16]
    F.G. Gustavson, “Some basic techniques for solving sparse systems of linear equations”, in: D.J. Rose and R.A. Willoughby, eds.,Sparse matrices and their applications (Plenum, New York, 1972) pp. 41–52.Google Scholar
  17. [17]
    P. Harris, “Pivot selection rules of the Devex LP code”,Mathematical Programming Study 4 (1975) 30–57.Google Scholar
  18. [18]
    E. Hellerman and D. Rarick, “The partitioned preassigned pivot procedure (P4)” in: D.J. Rose and R.A. Willoughby, eds.,Sparse matrices and their applications (Plenum, New York, 1972) pp. 67–76.Google Scholar
  19. [19]
    J.K. Ho, “Implementation and application of a nested decomposition algorithm”, in: W.W. White, ed.,Computers and mathematical programming (National Bureau of Standards, 1978) pp. 21–30.Google Scholar
  20. [20]
    J.K. Ho, E. Loute, Y. Smeers and E. van de Voort, “The use of decomposition techniques for large scale linear programming energy models”, in: A. Strub, ed.,Energy Policy, special issue on “Energy Models for the European Community”, (1979) 94–101.Google Scholar
  21. [21]
    J.K. Ho and E. Loute, “A comparative study of two methods for staircase linear programs”,ACM Transactions on Mathematical Software 6 (1980) 17–30.Google Scholar
  22. [22]
    IBM, “Mathematical Programming System Extended (MPSX) Read Communication Format (READCOMM) Program description manual”, SH20-0960-0 (January 1971).Google Scholar
  23. [23]
    IBM, “IBM Mathematical Programming System Extended/370 (MPSX/370) Program reference manual”, SH19-1095-3 (December 1979).Google Scholar
  24. [24]
    IBM, “IBM Mathematical Programming System. Extended/370 (MPSX/370) Logic manual”, (licensed material) LY19-1024-0 (January 1975).Google Scholar
  25. [25]
    J.E. Kalan, “Aspects of large-scale in core linear programming”, in:Proceedings of A.C.M. annual conference (1971) 304–313.Google Scholar
  26. [26]
    G.P. Kutcher, “On the decomposition price endogenous models”, in: L.M. Goreux and A.S. Manne, eds.,Multi-level planning: Case studies in Mexico (North-Holland, Amsterdam, 1973) pp. 499–519.Google Scholar
  27. [27]
    L.S. Lasdon,Optimization theory for large systems (McMillan, London, 1970).Google Scholar
  28. [28]
    H.M. Markowitz, “The elimination form of the inverse and its application to linear programming”,Management Science 3 (1957) 255–269.Google Scholar
  29. [29]
    W. Orchard—Hays,Advanced linear-programming computing techniques (McGraw—Hill, New York, 1968).Google Scholar
  30. [30]
    M. Simmonard,Programmation linéaire, Vols. 1 et 2 (Dunod, Paris, 1972).Google Scholar
  31. [31]
    L. Slate and K. Spielberg, “The extended control language of MPSX/370 and possible applications”,IBM Systems Journal 17 (1978) 64–81.Google Scholar
  32. [32]
    G. von Schiefer, “Lösung grosser linear Regionalplannungsprobleme mit der Methode von Dantzig und Wolfe”,Zeitschrift für Operations Research, 20 (1976) B1-B16.Google Scholar
  33. [33]
    H.P. Williams and A.C. Redwood, “A structured programming model in the food industry”,Operational Research Quarterly 25 (1974) 517–527.Google Scholar

Copyright information

© North-Holland Publishing Company 1981

Authors and Affiliations

  • James K. Ho
    • 1
  • Etienne Loute
    • 2
  1. 1.Applied Mathematics DepartmentBrookhaven National LaboratoryUptonUSA
  2. 2.C.O.R.E., Université Catholique de LouvainLouvain-la-NeuveBelgium

Personalised recommendations