Algorithms For Mixed Variational Inequalities With Applications

Variational inequality has become an important part of nonlinear analysis since the 1960s,which is one research hotspot of optimization theory.Variational inequality dates from related issues of mechanics,including resistance,free boundary problem,tangent problem,elastic plastic etc.Nowadays,variational inequality has become a general framework dealing with valid problems of many fields,such as applied mathematics,optimal control,mechanics,physics,nonlinear programming,economics,elastic and plastic,traffic,logistics,finance,etc.A useful and important generalization of variational inequalities is the mixed variational inequality containing a nonlinear term.On the base of mixed variational inequality and variant variational inequality,mixed variant variational inequality is proposed in this thesis.The projection method is limited due to applied to suggest iterative algorithms for solving mixed variant variational inequality except the presence of the nonlinear term in very special cases.Algorithms are created by combining resolvent operator technique with fixed point theory,whose convergence is proved.The validity and stability of algorithms is verified by numerical results.Differential variational inequality is newly developed , differential mixed variational inequality is a generation of differential variational inequality.Multiple steps Euler method is suggested solving the initial-value differential mixed variational inequality problem and the convergence of the method is analyzed.Variational inequality is widely applied in urban traffic,applications of the mixed variational inequalities in urban transportation management are depicted.Based on various variational inequality models of urban transportation network equilibrium and many problems of mixed traffic network,mixed variational inequality models under some charge system of urban traffic congestion are proposed.