Research On Relaxed Projection Algorithms For Variational Inequality Problems

Variational inequality problem(VIP)is a very important research field in the optimization problem.It has been widely used in signal processing and image reconstruction,system recognition,economic science,filtering design,automatic control and other fields.Since this problem was proposed,many numerical methods have been used to solve it,and more and more experts and scholars from home and abroad have paid extensive attention to the theory and method of this problem.Among these methods,the projection algorithm is very representative.Especially when the projection area is easy to calculate,the amount of calculation is very small;when constraints are simple,it also just takes up very little space.Therefore,based on this idea,this paper makes a further study.It is divided into four chapters,and its structure is arranged as follows:The first chapter mainly introduces the application background,research status and main works of this paper.The second chapter gives out the solution of the VIP of the modified double extragradient subgradient projection algorithm.The selection of step length rules was improved,the demand of function (8 is weak lower semicontinuous was also removed.The algorithm may require mapping F is Lipscitz continuous,but the algorithm does not require mapping Lipschitz constants,and Lipschitz constant in actual operation is difficult to obtain.As a result,the algorithm has expanded the scope of application to a certain extent,the numerical example is given to verify the effectiveness of the algorithm.The third chapter proposes a sequence projection algorithm for the VIP under multiple sets.When the feasible set of the VIP is the intersection of multiple closed convex sets,we design a sequence projection algorithm that projects onto each closed convex set to solve such problems.When the projection area is relatively simple,the sequence projection algorithm can meet the demand;when the projection area is relatively complex,it is difficult to obtain the projection on the closed convex set.Therefore,we put forward the relaxed sequence projection algorithm,which uses the subgradient half space containing the closed convex set to replace the original projection area,which can make the algorithm simple and effective.Finally,numerical examples are given to verify the feasibility of the algorithm.The fourth chapter summarizes the research content of this paper,and puts forward the direction of further research.