Generalized Ishikawa And Viscosity Approximation Iterative Algorithm And Its Application

Fixed point problem is the basis of optimization research,in this dissertation,two generalized iterative algorithms are proposed to solve the fixed point problem of nonexpansive mappings.One is the generalized Ishikawa iterative algorithm,the other is the generalized viscosity approximation iterative algorithm.The generalized essence is to generalize the sum of coefficients equal to 1 in Ishikawa iterative algorithm and viscosity approximation iterative algorithm to be less than or equal to 1.Compared with the classical case,the parameter selection of generalized iterative algorithm is more flexible.As an application,we apply the generalized Ishikawa iterative algorithm to the variational inequality problem,and the generalized viscosity approximation iterative algorithm to the constrained convex optimization problem and bilevel optimization problem.At the same time,this dissertation generalizes some existing algorithms and puts forward some problems that can be further studied.In the first chapter,we introduce the fixed point problem of non-expansive mapping and its research status at home and abroad.At the same time,we clarify the research content and arrangement of this paper.In chapter 3,we propose the weak convergence theorem of generalized Ishikawa iterative algorithm,prove its weak convergence and give a concrete parameter example.Then,the generalized Ishikawa iterative algorithm is applied to solving a class of variational inequalities,and the solution of the weak convergence of the algorithm to the variational inequality is proved.In chapter 4,we propose the strong convergence theorem of generalized viscosity approximation iterative algorithm,prove its strong convergence and give a concrete parameter example.Then,the generalized viscosity approximation iterative algorithm is applied to the constrained convex optimization problem and the bilevel optimization problem.The generalized viscosity gradient projection algorithm and the belevel generalized viscosity approximation algorithm are proposed respectively.Finally,it is proved that the algorithm converges strongly to the solution of the optimization problem.In chapter 5,we summarize and look forward to this paper.The main work and achievements of this paper are explained.At the same time,the shortcomings of this paper and the direction of improvement are put forward.