Solution of Large-Scale Multicommodity Network Flow Problems Via a Logarithmic Barrier Function Decomposition

  • Heinrich Lange
  • R. Kevin Wood
Conference paper
Part of the Operations Research Proceedings book series (ORP, volume 1990)


A new algorithm is presented using a logarithmic barrier function decomposition for the solution of the large-scale multicom-modity network flow problem. Placing the complicating joint capacity constraints into a logarithmic barrier term of the objective function creates a nonlinear mathematical program with linear network flow constraints which is solved by using the technique of restricted simplicial decomposition. Computational results on a network with 3,300 nodes and 10,400 arcs are reported for four, ten and 100 commodities.


Barrier Function Master Problem Network Flow Problem Transshipment Problem Logarithmic Barrier Function 
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.


Vorgestellt wird ein neuer Algorithmus zur Optimierung von Mehrproduktflüssen in großen bewerteten Kapazitäten-digraphen. Unter Verwendung einer logarithmischen Sperrfunktion werden die produktübergreifenden Kapazitätsbeschränkungen in die Zielfunktion aufgenommen. Das transformierte nichtlineare Optimierungsproblem mit linearen Nebenbedingungen wird durch die sogenannte „restricted simplicial decomposition“ gelöst. Über Rechnererfahrungen an einem Netzwerk mit 3.300 Knoten und 10.400 Kanten wird für vier, zehn und 100 Produkte berichtet.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. /l/.
    Bradley, G. H.,Brown, G. G., and Graves, G. W.: Design and Implementation of Large Scale Primal Transshipment Algorithms, Management Science, Vol. 24–1, 1–34 (1977)CrossRefGoogle Scholar
  2. /2/.
    Kennington, J. L., Helgason, R. V.: Algorithms for Network Programming, New York: John Wiley & Sons (1980)Google Scholar
  3. /3/.
    Staniec, C. J.: Solving the Multicommodity Transshipment Problem, Ph.D. Dissertation, Naval Postgraduate School, Monterey/CA (1987)Google Scholar
  4. /4/.
    Frisch, K. R.: The Logarithmic Potential Method of Convex Programming, unpublished manuscript, University Institute of Economics, Oslo (1955)Google Scholar
  5. /5/.
    Fiacco, A. V.; McCormick, G. P., Nonlinear Programming: Sequential Unconstrained Minimization Techniques, New York: John Wiley & Sons (1968)Google Scholar
  6. /6/.
    Gill, P. E.; Murray, W.; Saunders, M. A.; Tomlin, J.A.; Wright, M. H.: A Note on Nonlinear Approaches to Linear Programming, Systems Optimization Labratory Technical Report SOL 86–7, (1986)Google Scholar
  7. /7/.
    Karmarkar, N. K.: A New Polynomial-Time Algorithm for Linear Programming. Combinatorica, Vol.2, 373–395 (1984)CrossRefGoogle Scholar
  8. /8/.
    Megiddo, N.: Pathways to the Optimal Set in Linear Programming. Preliminary Report, IBM Almaden Research Center, San Jose/CA, and Tel Aviv University (1986)Google Scholar
  9. /9/.
    Wright, M. A.: Numerical Methods for Nonlinear Constrained Optimization, Ph.D. Dissertation, Stanford University/CA (1976)Google Scholar
  10. /10/.
    Hearn, D. W.; Lawphongpanich, S.; Ventura, J. A.: Restricted Simplicial Decomposition: Computation and Extensions. Mathematical Programming Study, Vol.31, 99–118 (1987)Google Scholar
  11. /11/.
    Frank, M.; P. Wolfe: An Algorithm for Quadratic Programming. Naval Research Logistics Quarterly,Vol.3, 95–110 (1956)CrossRefGoogle Scholar
  12. /12/.
    Geoffrion, A. M.: Elements of Large-scale Mathematical Programming. Management Science, Vol.16, 652–691 (1970)CrossRefGoogle Scholar
  13. /13/.
    Von Hohenbalken, B., A Finite Algorithm to Maximize Certain Pseudoconcave Functions on Polytopes. Mathematical Programming, Vol. 8, 189–206 (1975)CrossRefGoogle Scholar
  14. /14/.
    Holloway, C. A.: An Extension of the Frank and Wolfe Method of Feasible Directions. Mathematical Programming, 6/14–27 (1974)CrossRefGoogle Scholar
  15. /15/.
    Wolfe, P.: Methods for Nonlinear Programming. Recent Advances in Mathematical Programming, ed. Graves, R.L. and P. Wolfe (1963)Google Scholar
  16. /16/.
    McCormick, G. P.: Anti-Zig-Zagging by Bending. Management Science, Vol.15, 315–320 (1969)CrossRefGoogle Scholar
  17. /17/.
    Reklaitis, G. V.; Ravindran, A.; Ragsdell, K.M.: Engineering Optimization, New York: John Wiley & Sons (1983)Google Scholar
  18. /18/.
    Brown, G.G.; G.W. Graves, Design and Implementation of a Large-scale Optimization System. Presented at ORSA/TIMS Conference, Las Vegas, Nevada (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1992

Authors and Affiliations

  • Heinrich Lange
    • 1
  • R. Kevin Wood
    • 2
  1. 1.HermannsburgGermany
  2. 2.MontereyUSA

Personalised recommendations