On A Class Of Structure Monotone Inclusion And Its Application In Convex Optimization

Many problems in image restoration and image reconstruction can be expressed as convex optimization problems.In order to solve these convex optimization problems,under certain conditions,according to Fermat lemma,they can usually be transformed into monotone inclusion problems.Operator splitting algorithm is an important kind of iterative algorithm for monotone inclusion,including forward backward splitting algorithm,Douglas Rachford splitting algorithm And T'seng splitting algorithm.In particular,resolvent is the basic concept of various operator splitting algorithms,in which almost all of them have the calculation of resolvent.But for the resolvent of some composed operators,it is not easy to calculate.In this paper,we propose a fixed point iterative method to solve the resolvent of a class of composite operators.In addition,we study the monotone inclusion problem of finite sum.The existing operator splitting algorithm has some shortcomings in the process of solving,for example,it involves solving subproblems,resulting in low efficiency.Therefore,in this paper,we propose a completely splitting method to solve the monotone inclusion problem.Furthermore,we apply the results to the image denoising problem of impulse noise.The paper is divided into four chapters,as follows:In the first chapter,we introduce the background of optimization problem,monotone inclusion problem and the research status of its iterative algorithm.Then some symbols and definitions are given.Finally,the main contents of this paper are described.In the second chapter,we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators.First,we discuss several explicit solutions of this resolvent operator by taking into account additional constraints on the linear operator.Second,we propose a fixed point approach for computing this resolvent operator in a general case.Based on the Krasnoselskii-Mann algorithm for finding fixed points of non-expansive operators,we prove the strong convergence of the sequence generated by the proposed algorithm.As a consequence,we obtain an effective iterative algorithm for solving the scaled proximity operator of a convex function composed by a linear operator,which has wide applications in image restoration and image reconstruction problems.Furthermore,we propose and study iterative algorithms for studying the resolvent operator of a finite sum of maximally monotone operators as well as the proximal operator of a finite sum of proper,lower semi-continuous convex functions.In the third chapter,This chapter is concerned with the monotone inclusion involving the sum of a finite number of maximally monotone operators and the parallel sum of two maximally monotone operators with bounded linear operators.To solve this monotone inclusion,we first transform it into the formulation of the sum of three maximally monotone operators in a proper product space.Then we derive two efficient iterative algorithms,which combine the partial inverse method with the preconditioned Douglas-Rachford splitting algorithm and the preconditioned proximal point algorithm.Furthermore,we develop an iterative algorithm reply on the preconditioned Douglas-Rachford splitting algorithm without using the partial inverse method.We carefully analyze the theoretical convergence of the proposed algorithms.Finally,in order to demonstrate the effectiveness and efficiency of these algorithms,we conduct numerical experiments on a novel image denoising model for salt-and-pepper noise removal.Numerical results show the good performance of the proposed algorithms.The fourth chapter summarizes the full text and gives a prospect for future work.