Network equilibrium models and algorithms.

*(English)*Zbl 0864.90042
Ball, M. O. (ed.) et al., Network routing. Amsterdam: North-Holland. Handb. Oper. Res. Manage. Sci. 8, 485-550 (1995).

The general network equilibrium model (NEM) is concerned with determining flow patterns in a network where cost and demand functions governing the flows are nonlinear. The NEM has application to problems in transportation planning, spatial economics, water pipe networks and electrical networks. This chapter provides an overview of the models and algorithms for network equilibrium problems which have been developed over the past 20 years.

The first two sections concentrate on the basic NEM model, giving standard numerical examples from urban traffic assignment. The NEM is formulated as a variational inequality problem and the equivalence with nonlinear complementarily and fixed point formulations is given. Conditions for existence and uniqueness of solutions are then proven. Section 3 contains several optimization reformulations, especially for the case where the cost and demand functions are symmetric. The following two sections describe basic algorithms for symmetric and asymmetric versions of the NEM, including linear approximation, path equilibration, simplicial decomposition, projection and relaxation methods, and dual methods.

Section 6 contains specific examples of NEMs for spatial price equilibrium, water pipe and electrical networks. The final section contains reference notes, including a subsection on specialized topics not covered. The chapter concludes with references to 227 articles and texts in the field.

For the entire collection see [Zbl 0829.00010].

The first two sections concentrate on the basic NEM model, giving standard numerical examples from urban traffic assignment. The NEM is formulated as a variational inequality problem and the equivalence with nonlinear complementarily and fixed point formulations is given. Conditions for existence and uniqueness of solutions are then proven. Section 3 contains several optimization reformulations, especially for the case where the cost and demand functions are symmetric. The following two sections describe basic algorithms for symmetric and asymmetric versions of the NEM, including linear approximation, path equilibration, simplicial decomposition, projection and relaxation methods, and dual methods.

Section 6 contains specific examples of NEMs for spatial price equilibrium, water pipe and electrical networks. The final section contains reference notes, including a subsection on specialized topics not covered. The chapter concludes with references to 227 articles and texts in the field.

For the entire collection see [Zbl 0829.00010].

Reviewer: D.Hearn (Gainesville, U.S.A.)

##### MSC:

90B06 | Transportation, logistics and supply chain management |

90B10 | Deterministic network models in operations research |

90C35 | Programming involving graphs or networks |

49J52 | Nonsmooth analysis |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

91D25 | Spatial models in sociology |