The convergence of diagonalization algorithms for asymmetric network equilibrium problems

doi:10.1016/0191-2615(82)90007-8

Transportation Research Part B: Methodological

Volume 16, Issue 6, December 1982, Pages 477-483

https://doi.org/10.1016/0191-2615(82)90007-8 Get rights and content

Abstract

We provide a sufficient condition for the convergence of diagonalization algorithms for equilibrium traffic assignment problems with asymmetric Jacobian matrix B(v) of the link user cost mapping s(v) of the flow v. When $|I−D(v^{∗})^{−12} B(v^{∗}) D(v^{∗}^{−12} |_{2} < 1$ , where $D(v^{∗}) > 0$ is the diagonal of B(v^∗) and v^∗ is the equilibrium flow, we demonstrate a local convergence theorem for nonlinear cost functions. The implication of this result for practical applications of the model are outlined.

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One of the most commonly used approaches to solve asymmetric network assignment problems is the diagonalization method because of its easy implementation (de Cea & Fernández, 1993; Florian, 1977). The convergence of diagonalization algorithms for asymmetric network equilibrium problems has been explored by Florian & Spiess (1982), and they provided a sufficient condition for the convergence. The symmetric assignment problem in Eq. (42) has an equivalent convex optimization formulation:
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^☆: This research was supported by a grant from the Natural Science and Engineering Research Council of Canada.

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The convergence of diagonalization algorithms for asymmetric network equilibrium problems☆

Abstract

Comp. Oper. Res.

Transpn Res.

Transpn Res. B

Equilibria on a congested transportation network

SIAM J. on Algebraic and Discrete Methods

Methods for combining modal split and equilibrium assignment models

Transpn Sci.

Computation of market equilibria for policy analysis: the Project Independence Evaluation System Approach

Studies in the Economics of Transportation

An extended traffic assignment model with applications to two-way traffic

Transn Sci.

Decomposition algorithms for the general asymmetric traffic equilibrium problem

Traffic equilibrium and variational inequalities

Transpn Sci.

Solution Algorithms for Network Equilibrium Models with Asymmetric User Costs