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### Open Access Journals

JIMO

In the last decade, as calibrations of the classical traffic
equilibrium problems, various models of traffic equilibrium problems
with nonadditive route costs have been proposed. For solving such
models, this paper develops a self-adaptive projection-auxiliary
problem method for monotone variational inequality (VI) problems. It
first converts the original problem where the feasible set is the
intersection of a linear manifold and a simple set to an augmented
VI with simple set, which makes the projection easy to implement.
The self-adaptive strategy avoids the difficult task of choosing
`suitable' parameters, and leads to fast convergence. Under suitable
conditions, we prove the global convergence of the method. Some
preliminary computational results are presented to illustrate the
ability and efficiency of the method.

JIMO

Without the information of the origin-destination demand function
and users' valuation for travel time saving, the precise estimation
of the road tolls for various pricing schemes must go in a
trial-and-error manner, as suggested by [2] and
[15], and recently realized by [6, 7, 11, 22, 24]. For a trial of the tolls pattern, the
responses of the users can be observed and used to update the toll
pattern for the next trial. Since getting the responses of the users
is expensive, it is desirable to use the acquired information
exhaustively; That is, we need to make the method converge to an
approximate solution of the problem within as little number of
changes as possible.

In this paper, we propose to update the link tolls pattern in an improved manner, where the profit direction is the combination of two known directions. This combined manner makes the method more efficient than the method using solely one of them. We prove the global convergence of the method under suitable conditions as those in [6, 7, 24]. Some preliminary computational results are also reported.

In this paper, we propose to update the link tolls pattern in an improved manner, where the profit direction is the combination of two known directions. This combined manner makes the method more efficient than the method using solely one of them. We prove the global convergence of the method under suitable conditions as those in [6, 7, 24]. Some preliminary computational results are also reported.

JIMO

Recently, Han (Han D, Inexact operator splitting
methods with self-adaptive strategy for
variational inequality problems,

*Journal of Optimization Theory and Applications***132**, 227-243 (2007)) proposed an inexact operator splitting method for solving variational inequality problems. It has advantage over the classical operator splitting method of Douglas-Peaceman-Rachford-Varga operator splitting methods (DPRV methods) and some of their variants, since it adopts a very flexible approximate rule in solving the subproblem in each iteration. However, its convergence is established under somewhat stringent condition that the underlying mapping $F$ is strongly monotone. In this paper, we mainly establish the global convergence of the method under weaker condition that the underlying mapping $F$ is monotone, which extends the fields of applications of the method relatively. Some numerical results are also presented to illustrate the method.## Year of publication

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