Research On Some Methods Of Variational Inequalities And Their Applications To Transportation Management

As the continuous development of interdisciplinary,variational inequality(VI)problem plays an increasingly important role in many fields.This dissertation mainly considers the road congestion pricing problem of traffic manage-ment.Its mathematical model can be characterized as a class of variational inequalities in which part of the underlying mappings is unknown(demand function is unknown).Moreover,consid-ering the capacity of link flows,we further consider the road congestion pricing problem with capacity constraints.Its mathematical model can be regarded as such a category of variational inequalities with unknown mappings and linear constraints.Since variational inequalities with linear constraints have wide applications,we also consider more general variational inequalities with nonlinear constraints.Firstly,operator splitting methods can be applied to solve VI problem with partially un-known mappings.However,the recently developed methods require restrictive conditions such as strong monotonicity,etc.On the one hand,it excludes many interesting applications.On the other hand,concrete issues in the transportation system normally do not satisfy the re-quirement.Therefore,based on existing methods,we propose a new operator splitting method by adding the regularization term similar to a proximal point algorithm regularization term.Under the mild condition that the underlying mapping is monotone,we prove the global con-vergence of the proposed method.Meanwhile,we get O(1/t)and O(1/t)convergence rate in non-ergodic sense.We also report some preliminary numerical results which show that the new algorithm is effective.Secondly,for VI problem with linear constraints,the prediction and correction method is a classic iteration method.Applying the method to the specific problem,sub-problem's solution can be obtained by observing the response of drivers.Note that the cost of the observation is very expensive.Especially when the iterative vector is far from the solution set,there is little justification to get the exact solution of sub-problem.So,the sub-problem can be solved by adopting an inexact strategy.Under the same condition,the global convergence of the proposed method is proved.Some numerical examples are presented to illustrate the efficiency of the inexact strategy.Finally,we propose a new inexact smoothing Newton method for solving general ? prob-lem with nonlinear constraints.Both global and local quadratic convergence are established under mild conditions.Some numerical results show that the proposed method is stable and effective.