Abstract
The purpose of the note is to look at the problem of biproportional matrix balancing when upper bounds are imposed on the matrix elements. This problem can be formulated as a convex minimization problem. Using the Kuhn-Tucker optimality conditions the functional form of the resulting model is derived. The dual formulation of the problem is derived and it is shown how it can be solved by a cyclic coordinate descent method. This leads to the proposal of an efficient solution algorithm.
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References
Bregman, L. (1967). “The Relaxation Method of Finding the Common Point of Convex Sets and its Application to the Solution of Problems in Convex Programming,” U.S.S.R. Computational Math. Mathematical Phys. 7, 200–217.
Chabini, I. and Florian, M. (1995). “An Entropy Based Primal-Dual Algorithm for Convex and Linear Cost Transportation Problems,” Centre de recherche sur les transports, Université de Montréal, Publication 963 (CRT #95–17).
Evans, S.P. and Kirby, H.R. (1974). “A Three-Dimensional Furness Procedure for Calibrating Gravity Models,” Transportation Research, 8, 105–122.
Furness, K.P. (1970). “Time Function Interaction,” Traffic Engineering and Control Vol 7, No 7, 19–36.
INRO Consultants Inc. (1997). EMME/2 User’s Manual.
Lamond, B. and Stewart, N.F. (1981). “Bregman’s Balancing Method,” Transportation Research, 15B, 239–248.
Luenberger, D.G. (1984). Linear and Nonlinear Programming. Second Edition, Addison-Wesley.
Murchland, J. (1977). The Multi-proportional Problem. Univerity College London, research note JDM 263.
Spiess, H. (1984). Contributions à la théorie et aux outils de planification de réseaux de transport urbain. Ph.D. thesis, Département d’informatique et de recherche opérationnelle, Publication 382, Centre de recherche sur les transports, Université de Montréal.
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Spiess, H. (2002). Biproportional Matrix Balancing with Upper Bounds. In: Gendreau, M., Marcotte, P. (eds) Transportation and Network Analysis: Current Trends. Applied Optimization, vol 63. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6871-8_15
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DOI: https://doi.org/10.1007/978-1-4757-6871-8_15
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5212-7
Online ISBN: 978-1-4757-6871-8
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