Abstract
A modification of the column generation operation in Dantzig—Wolfe decomposition is suggested. Instead of the usual procedure of solving one or more subproblems at each major iteration, it is shown how the subproblems may be solved parametrically in such a way as to maximize the immediate improvement in the value of objective in the “master problem”, rather than to maximize the “reduced profit” of the entering column. The parametric problem is shown to involve the maximization of a piece-wise linear concave function of a single variable. It is hoped that in some cases the use of the suggested procedure may improve the slow rates of convergence common in decomposition algorithms.
This is a preview of subscription content, log in via an institution to check access.
References
E.M.L. Beal, P.A.B. Hughes and R.E. Small, “Experiences in using a decomposition program”,Computer Journal 8 (1965) 13–18.
P. Broise, P. Huard and J. Sentenac, “Decomposition des programmes mathematiques”, in:Monographies de recherche operationnelle, No. 6 (Dunod, Paris, 1967).
G.B. Dantzig and P. Wolfe, “Decomposition principle for linear programs”,Operations Research 8 (1960) 101–111.
E. Hellerman, “Large scale linear programs: Theory and computation”, in:Operations research and the design of management information systems, Ed. J.F. Pierce (Tech. Assn. of the Pulp and Paper Industry, New York, 1967) pp. 60–75.
L. Lasdon,Optimization theory for large systems (Macmillan, New York, 1970).
W. Orchard-Hays,Advanced L.P. computing techniques (McGraw-Hill, New York, 1968).
W. Orchard-Hays, “Practical problems in L.P. decomposition and a standardized phase I decomposition as a tool for solving large scale problems”, in:Decomposition of largescale problems,
M. Simmonard,Linear programming (Prentice Hall, Englewood Cliffs, N.J., 1966).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Swoveland, C. A note on column generation in Dantzig—Wolfe decomposition algorithms. Mathematical Programming 6, 365–370 (1974). https://doi.org/10.1007/BF01580252
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580252