# A variational model for equilibrium problems in a traffic network

Giandomenico Mastroeni; Massimo Pappalardo

RAIRO - Operations Research (2010)

- 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 38.1 (2010): 3-12. <http://eudml.org/doc/105302>.

@article{Mastroeni2010,

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},

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

language = {eng},

month = {3},

number = {1},

pages = {3-12},

publisher = {EDP Sciences},

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

url = {http://eudml.org/doc/105302},

volume = {38},

year = {2010},

}

TY - JOUR

AU - Mastroeni, Giandomenico

AU - Pappalardo, Massimo

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

JO - RAIRO - Operations Research

DA - 2010/3//

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.; network flows; equilibrium problems; transportation problems

UR - http://eudml.org/doc/105302

ER -

## References

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- S. Dafermos, Traffic equilibria and variational inequalities. Math. Program.26 (1980) 40-47.
- P. Ferrari, Capacity Constraints in Urban Transport Networks. Transpn. Res. B31 (1997) 291-301.
- 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.
- 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.
- A. Maugeri, W. Oettli and D. Sclager, A flexible form of Wardrop principle for traffic equilibria with side constraints. Rendiconti del Circolo Matematico di Palermo48 (1997) 185-193.
- 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).
- M. Patriksson, Nonlinear Programming and Variational Inequality Problems. Kluwer Academic Publishers, Dordrecht (1999).
- 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.

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