# A variational model for equilibrium problems in a traffic network

Giandomenico Mastroeni; Massimo Pappalardo

RAIRO - Operations Research - Recherche Opérationnelle (2004)

- Volume: 38, Issue: 1, page 3-12
- ISSN: 0399-0559

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topMastroeni, Giandomenico, and Pappalardo, Massimo. "A variational model for equilibrium problems in a traffic network." RAIRO - Operations Research - Recherche Opérationnelle 38.1 (2004): 3-12. <http://eudml.org/doc/245887>.

@article{Mastroeni2004,

abstract = {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.},

author = {Mastroeni, Giandomenico, Pappalardo, Massimo},

journal = {RAIRO - Operations Research - Recherche Opérationnelle},

keywords = {network flows; variational inequalities; equilibrium problems; traffic problems; transportation problems},

language = {eng},

number = {1},

pages = {3-12},

publisher = {EDP-Sciences},

title = {A variational model for equilibrium problems in a traffic network},

url = {http://eudml.org/doc/245887},

volume = {38},

year = {2004},

}

TY - JOUR

AU - Mastroeni, Giandomenico

AU - Pappalardo, Massimo

TI - A variational model for equilibrium problems in a traffic network

JO - RAIRO - Operations Research - Recherche Opérationnelle

PY - 2004

PB - EDP-Sciences

VL - 38

IS - 1

SP - 3

EP - 12

AB - 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.

LA - eng

KW - network flows; variational inequalities; equilibrium problems; traffic problems; transportation problems

UR - http://eudml.org/doc/245887

ER -

## References

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- [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). Zbl1129.90304MR2043480
- [8] M. Patriksson, Nonlinear Programming and Variational Inequality Problems. Kluwer Academic Publishers, Dordrecht (1999). Zbl0913.65058MR1673631
- [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. Zbl0847.49012MR1331417

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