Two Primal Dual Algorithms For Solving Monotone Inclusion Involving Parallel-sums And Compositions With Linear Operators

Many problems,arising from machine learning,signal processing,can be casted into a inclusion problem for finding a zero of the sum of two monotone operators.The forward-backward splitting algorithm and the Douglas-Rachford splitting algorithm are very typical methods for solving such a problem.The monotone inclusion problem with parallel and linear operators is a more general form of the zeros of two monotone operators.This article consists of four parts.In the first chapter,firstly,we introduce the problem of finding a zero of the sum of two monotone operators,by forward-backward splitting algorithm,and its research status.Then,it is the research status of the DR splitting algorithm and the monotonous inclusion involving parallel-sums and compositions with linear operators.Finally,the basic knowledge,lemmas and propositions to be used in this paper are given.In the second chapter,we consider the monotone inclusion involving parallel-sums and compositions with linear operators,on the basis of the existing modified FB splitting algorithm,and give a kind of FB type primal-dual algorithm for solving such a problem,and give the proof of the strong convergence of the algorithm.In the third chapter,firstly,we give a DR type splitting algorithm for finding a zero point of two maximal monotone operators,and the strong convergence of the generation sequence is proved,and then we give a primal-dual algorithm of DR type,for the monotonous inclusion involving parallel-sums and compositions with linear operators.Finally,we show the strong convergence of the algorithm.In the fourth chapter,we give a practical application of the research questions.