A variational model for equilibrium problems in a traffic network
RAIRO - Operations Research - Recherche Opérationnelle, Tome 38 (2004) no. 1, pp. 3-12.

We propose a variational model for one of the most important problems in traffic networks, namely, the network equilibrium flow that is, traditionally in the context of operations research, characterized by minimum cost flow. This model has the peculiarity of being formulated by means of a suitable variational inequality (VI) and its solution is called “equilibrium”. This model becomes a minimum cost model when the cost function is separable or, more general, when the jacobian of the cost operator is symmetric; in such cases a functional representing the total network utility exists. In fact in these cases we can write the first order optimality conditions which turn out to be a VI. In the other situations (i.e., when global utility functional does not exist), which occur much more often in the real problems, we can study the network by looking for equilibrium solutions instead of minimum points. The Lagrangean approach to the study of the VI allows us to introduce dual variables, associated to the constraints of the feasible set, which may receive interesting interpretations in terms of potentials associated to the arcs and the nodes of the network. This interpretation is an extension and generalization of the classic Bellman conditions. Finally, we deepen the analysis of the networks having capacity constraints.

DOI : 10.1051/ro:2004009
Classification : 90B10
Mots-clés : network flows, variational inequalities, equilibrium problems, traffic problems, transportation problems 
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Mastroeni, Giandomenico; Pappalardo, Massimo. A variational model for equilibrium problems in a traffic network. RAIRO - Operations Research - Recherche Opérationnelle, Tome 38 (2004) no. 1, pp. 3-12. doi : 10.1051/ro:2004009. http://archive.numdam.org/articles/10.1051/ro:2004009/

[1] M. Castellani, J.M. Jama and G. Mastroeni, Duality relations for variational inequalities with applications to Network Flows. Rend. Circ. Mat. Palermo 48 (1997) 39-55. | MR | Zbl

[2] S. Dafermos, Traffic equilibria and variational inequalities. Math. Program. 26 (1980) 40-47. | Zbl

[3] P. Ferrari, Capacity Constraints in Urban Transport Networks. Transpn. Res. B 31 (1997) 291-301.

[4] P.T. Harker and J.S. Pang, Finite-Dimensional variational inequalities and nonlinear complementarity problem: a survey of theory, algorithms and applications. Math. Program. 48 (1990) 161-220. | Zbl

[5] T. Larsson and M. Patriksson, On side constrained models for traffic equilibria, in Variational Inequalities and Network Equilibrium Problems, edited by F. Giannessi and A. Maugeri. Plenum Publishing, New York (1995) 169-179. | MR | Zbl

[6] A. Maugeri, W. Oettli and D. Sclager, A flexible form of Wardrop principle for traffic equilibria with side constraints. Rendiconti del Circolo Matematico di Palermo 48 (1997) 185-193. | MR | Zbl

[7] M. Pappalardo and M. Passacantando, Equilibrium concepts in transportation networks: generalized Wardrop conditions and variational formulations, to appear in Equilibrium problems: nonsmooth optimization and variational inequality models, edited by P. Daniele, A. Maugeri and F. Giannessi. Kluwer, Dordrecht (2003). | MR | Zbl

[8] M. Patriksson, Nonlinear Programming and Variational Inequality Problems. Kluwer Academic Publishers, Dordrecht (1999). | MR | Zbl

[9] R.T. Rockafellar, Monotone relations and network equilibrium, in Variational Inequalities and Network Equilibrium Problems, edited by F. Giannessi and A. Maugeri. Plenum Publishing, New York (1995) 271-288. | MR | Zbl

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