On The Convergence Of Relaxed Inertial Forward-reflected-backward Splitting Algorithms For Monotone Inclusions

The monotone inclusion is an important problem in the field of optimization.Under some conditions,convex optimization problems,variational inequality problems and saddle point problems can be transformed into the monotone inclusions.As an important algorithm for solving monotone inclusions,splitting algorithm was studied by many authors.Scholars modified this algorithm from different aspects by weakening the conditions or introducing acceleration techniques.In 2019,Malitsky and Tam proposed the relaxed inertial forward-reflected-backward splitting algorithm by introducing relaxation method and inertial technique,but the iterative step and inertia term of this algorithm were both fixed.When the iterative step and the inertia term are variable,it will speed up the algorithm.In this dissertation,based on the relaxed inertial forward-reflected-backward splitting algorithm,we modify the iterative step and the inertial term respectively,and propose two modified relaxed inertial forward-reflected-backward splitting algorithms for solving classical monotone inclusions.We establish the weakly convergence of the modified relaxed inertial forward-reflected-backward splitting algorithms without cocoercivity.Furthermore,we show the convergence rate of the algorithms under the mild conditions.This dissertation consists of the following five chapters:In chapter 1,we introduce the research background of splitting algorithm in Hilbert space at home and abroad and the main work of this dissertation.In chapter 2,we briefly describe the preliminary knowledge used in this dissertation.In chapter 3,we generalize the fixed step size of relaxed inertial forward-reflected-backward splitting algorithm proposed by Malitsky and Tam to variable coefficient.We analyse the convergence of the relaxed inertial forward-reflected-backward splitting algorithm with variable step,under the premise of α(28)0.We establish the weakly convergence of the sequence generated by the algorithm without cocoercivity,and show the convergence rate of the algorithm.In chapter 4,based on the relaxed inertial forward-reflected-backward splitting algorithm,we modify the algorithm with variable coefficient of the inertial term,propose the relaxed inertial forward-reflected-backward splitting algorithm with variable coefficient.Under the mild assumptions,we establish the weakly convergence of the sequence generated by the modified algorithm,and show the convergence rate of the new algorithm.In chapter 5,we summarize the full dissertation,explain the main conclusions and the future trends.