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Multi-Class User Equilibria under Social Marginal Cost Pricing
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
2002 (English)In: Operations Research 2002, 2002, 174-179 p.Conference paper, Published paper (Other academic)
Abstract [en]

In the congested cities of today, congestion pricing is a tempting alternative. With a single user class, already Beckmann et al. showed that ``system optimal'' traffic flows can be achieved by social marginal cost (SMC) pricing where users have to pay for the delays the incur on others. However different user classes can have widly differing time values. Hence, when introducing tolls, one should consider multi-class user equilibria, where the classes have different time values. In the single class case, the equilibrium conditions can be viewn as optimality conditions of an equivalent optimization problem. In the multi-class case, however, netter claims that this is not possible. We show that, depending on the formulation, the multi-class SMC-pricing equilibrium problem (with different time values) can be stated either as an asymmetric or as a symmetric equilibrium problem. In the latter case, the corresponding optimization problems is in general non-convex. For this non-convex problem, we devise descent methods of Frank-Wolfe type. We apply the methods and study a synthetic case based on Sioux Falls.

Place, publisher, year, edition, pages
2002. 174-179 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-14438ISBN: 978-3-540-00387-8 (print)OAI: oai:DiVA.org:liu-14438DiVA: diva2:23505
Available from: 2007-04-27 Created: 2007-04-27 Last updated: 2009-05-11
In thesis
1. Feasible Direction Methods for Constrained Nonlinear Optimization: Suggestions for Improvements
Open this publication in new window or tab >>Feasible Direction Methods for Constrained Nonlinear Optimization: Suggestions for Improvements
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns the development of novel feasible direction type algorithms for constrained nonlinear optimization. The new algorithms are based upon enhancements of the search direction determination and the line search steps.

The Frank-Wolfe method is popular for solving certain structured linearly constrained nonlinear problems, although its rate of convergence is often poor. We develop improved Frank--Wolfe type algorithms based on conjugate directions. In the conjugate direction Frank-Wolfe method a line search is performed along a direction which is conjugate to the previous one with respect to the Hessian matrix of the objective. A further refinement of this method is derived by applying conjugation with respect to the last two directions, instead of only the last one.

The new methods are applied to the single-class user traffic equilibrium problem, the multi-class user traffic equilibrium problem under social marginal cost pricing, and the stochastic transportation problem. In a limited set of computational tests the algorithms turn out to be quite efficient. Additionally, a feasible direction method with multi-dimensional search for the stochastic transportation problem is developed.

We also derive a novel sequential linear programming algorithm for general constrained nonlinear optimization problems, with the intention of being able to attack problems with large numbers of variables and constraints. The algorithm is based on inner approximations of both the primal and the dual spaces, which yields a method combining column and constraint generation in the primal space.

Place, publisher, year, edition, pages
Matematiska institutionen, 2007. 29 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1095
Keyword
constrained nonlinear optimization, feasible direction methods, conjugate directions, traffic equilibrium problem, sequential linear programming algorithm, stochastic transportation problem
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-8811 (URN)978-91-85715-11-4 (ISBN)
Public defence
2007-05-25, Alan Turing, Hus E, Campus Valla, Linköping University, Linköping, 10:15 (English)
Opponent
Supervisors
Note
The articles are note published due to copyright rextrictions.Available from: 2007-04-27 Created: 2007-04-27 Last updated: 2015-01-14

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Lindberg, Per OlovDaneva, Maria

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