JIMO
Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities
Gang Qian Deren Han Lingling Xu Hai Yang
Journal of Industrial & Management Optimization 2013, 9(1): 255-274 doi: 10.3934/jimo.2013.9.255
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.
keywords: auxiliary problem principle self-adaptive strategy. projection method Traffic equilibrium and nonadditive cost variational inequality problem
JIMO
Global convergence of an inexact operator splitting method for monotone variational inequalities
Zhili Ge Gang Qian Deren Han
Journal of Industrial & Management Optimization 2011, 7(4): 1013-1026 doi: 10.3934/jimo.2011.7.1013
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.
keywords: self-adaptive strategy. Monotone variational inequalities operator splitting methods approximate rules
JIMO
Congestion control with pricing in the absence of demand and cost functions: An improved trial and error method
Gang Qian Deren Han Hongjin He
Journal of Industrial & Management Optimization 2010, 6(1): 103-121 doi: 10.3934/jimo.2010.6.103
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.
keywords: unknown mappings. transportation network variational inequality problems trial-and-error Congestion pricing

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