Projection Algorithm For Quasi - Variational Inequality Problems

Quasi-variational inequality problem, variational inequality problem and nonlinear complementarity problem are important research problems in the field of optimization. They are widely used in economic, engineering, optimization, control and other fields. Nash equilibrium problem can be transformed into a variational inequality problem, and the generalized Nash equilibrium problem can be transformed into a quasi-variational inequality problem. Thus the effective numerical solution of the quasi-variational inequality problem is worth researching. We study the projection algorithm for quasi-variational inequality. The full text is divided into three chapters.The first chapter is an introduction. We state the definition of the quasi variational inequality problem, its research status and the main research work of this paper.In the second chapter, we designed a projection algorithm for solving the quasi-variational inequality problem with co-coercive. The prediction step is obtained without using a linesearch and the steplength of correction step is not fixed. At the end of this chapter, we give three examples to illustrate the feasibility and effectively of this algorithm.In the third chapter, we study the construction technique of hyperplane. The hyperplane is applied to two kinds of classical projection algorithms:The first projection algorithm contains a prediction step and correction step. So we need calculate at least twice projection at each iteration in this algorithm. We introduced hyperplane into the algorithm, which is easier to calculate the projection. The convergence of the algorithm is proved under the normal assumptions.The second is a hybrid projection algorithm, which only one projection is done to get a trial point and where the next point is obtained by a linesearch between this trial point and the current iterate. We consider the case when the projection onto the feasible set is not easy to calculate. Moreover, in the correction step we propose to combine the direction defined by the trial point and the current point with one of the three well-known directions used in the literature. The convergence is obtained under a general property satisfied by all of these three directions. We introduce the hyperplane into the hybrid projection algorithm. This improves the application range and computational efficiency of the algorithm. The convergence of the algorithm is proved under the normal assumptions.