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Beiträge zur Dekomposition von linearen Programmen

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Zusammenfassung

Das Ziel dieser Arbeit besteht vor allem darin, die Zusammenhänge der Dekompositionsmethoden für lineare Programme vonDantzig/Wolfe, Abadie/Williams, Benders, Rosen usw. aufzusuchen. In Kapitel I und III werden diese Methoden mehr oder weniger ausführlich behandelt. Für einige Sätze werden neue Beweise angegeben. In Kapitel II entwickeln wir eine neue Dekompositionsmethode, die aus der Kombination des primaldualen Verfahrens mit derDantzig/Wolfeschen Dekomposition entsteht. Der Idee vonCharnes/Cooper folgend benützen wir „dyadische Transformationen“ in Kapitel IV, um mehrstufige und Transportprobleme durch Dekomposition zu lösen. Schließlich werden in Kapitel V Überlicke über Anwendungen, Erweiterungen für nichtlineare Programmierung und einige Ausblicke gegeben.

Summary

In this paper we are concerned with the decomposition methods ofDantzig/Wolfe, Abadie/Williams, Benders andRosen, particularly to show the relations among them. A new method, the “primal-dual decomposition method” is developed. FollowingCharnes/Cooper, “dyadic transformations” are used to solve various linear programs by decomposition. A survey of extensions and applications is also given.

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Literaturverzeichnis

  1. Abadie, J. M.: Le principe de décomposition de Dantzig et Wolfe. Revue Française de Rech. Oper. 4 (1960), S. 93–115.

    Google Scholar 

  2. Abadie, J. M. undA. C. Williams: Dual and parametric methods in decomposition. In [35] (1963), S. 149–158.

  3. Balas, E.: Une méthode de décomposition quasi-primale-duale pour de programmes linéaires. I-Variante. C. R. Acad. Sci. Paris, 261 (1965), 4. Oct.

    Google Scholar 

  4. ——: idem.Une méthode de décomposition quasi-primale-duale pour de programmes linéaires. II-Variante. C. R. Acad. Sci. Paris, 261 (1965), 18. Oct.

    Google Scholar 

  5. Baumol, W. J. undF. Tibor: Decomposition, pricing for decentralisation and external economies. Manage. Sci. 11 (1964), S. 1–32.

    Google Scholar 

  6. Beale, E. M. L.: The Simplex method using pseudo-basic variables for structured linear prog. problems. In [35] (1963), S. 133–148.

  7. Beale, E. M. L., Hughes, P. A. B. undR. E. Small: Experiences in using a decomposition prog. The Computer J. 8 (1965), S. 13–18.

    Google Scholar 

  8. Benders, J. F.: Partition in math. prog. Diss. Utrecht 1960.

    Google Scholar 

  9. ——: Partitioning procedures for solving mixed-variables prog. problems. Num. Math. 4 (1962), S. 238–252.

    Google Scholar 

  10. Boot, J. C. G.: Quadratic prog. North-Holland Publ. Co. 1964.

  11. Charnes, A. undW. W. Cooper: Manage. models and industrial appl. of linear prog. Wiley, Vols 1 & 2, 1961.

  12. Černikov, S. N.: Contractions of systems of linear inequalities. Soviet Math. (Doklady) 1 (1960), S. 296–299.

    Google Scholar 

  13. ——: The solution of linear prog. problems by elimination of unknowns. Soviet Math. (Doklady) 2 (1961), S. 1099–1103.

    Google Scholar 

  14. ——: Contractions of finite systems of linear inequalities. Soviet Math. (Doklady) 4 (1963), S. 1520.

    Google Scholar 

  15. Dantzig, G. B.: Optimal solution of a dynamic Leontief model with substitution. Econometrica 23 (1955), S. 295–302.

    Google Scholar 

  16. ——: Upper bounds, secondary constraints, and block triangularity. Econometrica 23 (1955), S. 174–183.

    Google Scholar 

  17. Dantzig, G. B., Ford, L. R. undD. R. Fulkerson: A primal dual algorithm for linear prog. In [45] (1965), S. 171–181.

  18. Dantzig, G. B.: On the status of multistage linear prog. problems. Manage. Sci. 6 (1960), S. 53–72.

    Google Scholar 

  19. Dantzig, G. B. undP. Wolfe: Decomposition principle for linear prog. Oper. Res. 8 (1960), S. 101–111.

    Google Scholar 

  20. Dantzig, G. B.: General convex objective functions. InArrow/Karlin/Suppes: Math. methods in the social sciences, Stanford, 1960, Ch. 10.

  21. Dantzig, G. B. undM. B. Shapiro: Solving the chemical equilibrium problem using the decomp. principle. Rand Corp. P-2056, 1960, 10. Aug.

  22. Dantzig, G. B. undP. Wolfe: The decomp. algorithm for linear prog. Econometrica 29 (1961), S. 767–778.

    Google Scholar 

  23. Dantzig, G. B.: Compact basis triangularization for the simplex method. In [35] (1963), S. 125–132.

  24. --: Linear prog. and extentions. Princeton, 1963.

  25. Desport, Moustacchi undRaiman: Modèle expérimental de planification. J. Math. des Prog. Econom. Sofro, 1963, Nov.

  26. Ford, L. R. undD. R. Fulkerson: A simple algorithm for finding maximal network flows and an appl. to the Hitchcock problem. Rand Corp. P-743, 1955.

  27. ——: A suggested computation for maximal multicommodity network flows. Manage. Sci. 5 (1958), S. 97–101.

    Google Scholar 

  28. Gale, D.: The theory of linear economic models. McGraw Hill, 1960.

  29. Gass, S. I.: Linear prog.: methods and appl. McGraw Hill, 1958.

  30. Gauthier, J. M. undF. Genuys: Expériences sur les principe de décomposition des prog. linéaires. 1er Congress de l'Afcal, 1960.

  31. Gauthier, J. M.: Le principe de décomposition deDantzig etWolfe. Math. de prog. econom. Dunod, 1961.

  32. ——: Paramétrisation de la fonction econom. d'un prog. linéaires. Revue Franç. de Rech. Oper. 6 (1962), S. 5–20.

    Google Scholar 

  33. Gomory, R. E. undT. C. Hu: An appl. of generalized linear prog. to network flows. J. Siam 10 (1962), S. 260–283.

    Google Scholar 

  34. ——: Synthesis of a communication network. J. Siam 12 (1964), S. 348–365.

    Google Scholar 

  35. Gomory, R. E.: Large and nonconvex problems in linear prog. Proc. Symp. Appl. Math. 15 (1963), S. 125–139.

    Google Scholar 

  36. Graves, R. L. undP. Wolfe: Recent advances in math. prog. McGraw Hill, 1963.

  37. Hadley, G.: Linear prog. Addison Wesley, 1963.

  38. Harvey, R. P.: The decomp. principle for linear prog. International J. of Comp. Math. 1 (1964), S. 20–35.

    Google Scholar 

  39. Heesterman, A. R. G. undJ. Sandee: Special simplex algorithm for linked problems. Manage. Sci. 11 (1965), S. 420–428.

    Google Scholar 

  40. Huard, P.: Appl. du principe de décomp. aux prog. non linéaires Note Electr. de France, 1963, HR-5476/3.

  41. Jaeger, A. undB. Mond: On direct sums and tensor products of linear prog. Z. Wahrscheinlichkeit. 3 (1964), S. 19–31.

    Google Scholar 

  42. Kornai, J. undT. Lipták: Two level planning. Econometrica 33 (1965), S. 141–169.

    Google Scholar 

  43. Kreko, B.: Lehrbuch der linearen Optimierung. VEB Deutscher Verlag, 1964.

  44. Krelle, W. undH. P. Künzi: Lineare Prog. Industrielle Organisation, Zürich, 1958.

  45. Kron, G.: Diakoptics. The piecewise solution of large scale systems, I, II, III, IV, V, VI. Electrical J. 159 (1957), July–Dez.

  46. Kuhn, H. W. undA. W. Tucker: Linear inequalities and related systems. Princeton, 1956.

  47. Künzi, H. P. undW. Krelle: Nichtlineare Prog. Springer, 1962.

  48. Mond, B.: On direct sums and tensor products of nonlinear prog. Z. Wahrscheinlichkeit. 4 (1965), S. 27–39.

    Google Scholar 

  49. Müller-Merbach, H.: Das Verfahren der direkten Dekomposition in der linearen Planungsrechnung. Ablauf- und Planungsforschung 6 (1965), S. 306–322.

    Google Scholar 

  50. Narayamurthy, S. G.: A method for piecewise solution of a structured linear prog. J. Math. Anal. Appl. 10 (1965), S. 70–99.

    Google Scholar 

  51. Nemhauser, G.L.: Decomposition of linear prog. by dynamic prog. Naval Res. Log. Quart. 11 (1964), S. 191–195.

    Google Scholar 

  52. Pigot, D.: Double décomposition d'un prog. linéaire. Symp. Prog. London, 1964.

  53. Rosen, J. B.: Partition prog. Notices AMS 7 (1962), S. 718–719.

    Google Scholar 

  54. --: Convex partition prog. In [35] (1963), S. 159–176.

  55. ——: Primal partition prog. for block diagonal matrices. Num. Math. 6 (1964), S. 250–260.

    Google Scholar 

  56. ——: Solution of nonlinear prog. by partitioning. Manage. Sci. 10 (1963/64), S. 160–173.

    Google Scholar 

  57. Sanders, J. L.: A nonlinear decomposition principle. Oper. Res. 13 (1965), S. 266–271.

    Google Scholar 

  58. Simonnard, M.: Programmation linéaire. Dunod, 1962.

  59. Steward, D. V.: An approach to techniques for the analysis of the structure of large systems of equations. SIAM Review 4 (1962), S. 321–342.

    Google Scholar 

  60. Vajda, S.: Math. prog. Addison Wesley, 1961.

  61. Whinston, A.: A decomposition algorithm for quadratic prog. Cowles Found. Disc. Paper 172 (1964), 25. June.

  62. Williams, A.: A treatment of transportation problems by decomposition. J. SIAM 10 (1962), S. 35–48.

    Google Scholar 

  63. ——: A stochastic transportation problem. Oper. Res. 11 (1963), S. 759–770.

    Google Scholar 

  64. Wolfe, P.: Some simplex like nonlinear prog. procedures. Oper. Res. 10 (1962), S. 437–447.

    Google Scholar 

  65. Wolfe, P. undG. B. Dantzig: Linear prog. in aMarkov chain. Oper. Res. 10 (1962), S. 702–710.

    Google Scholar 

  66. Zoutendijk, G.: Methods of feasible directions. Elvesier, 1960.

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Authors and Affiliations

  1. Mathematisches Institut und Rechenzentrum der Universität Zürich, Rämistrße 71, 8006, Zürich

    S. T. Tan

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Diese Arbeit wurde von der Philosophischen Fakultät II der Universität Zürich als Dissertation angenommen.

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Tan, S.T. Beiträge zur Dekomposition von linearen Programmen. Unternehmensforschung Operations Research 10, 168–189 (1966). https://doi.org/10.1007/BF01920940

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