Reaserch On Four-operator Splitting Algorithms For Solving Monotone Inclusions And Its Applications

Many problems in signal and image processing,economic management and machine learning can be abstracted as convex optimization problems.According to the first-order optimality condition,these problems can usually be transformed into monotone inclusion problems.The operator splitting algorithm is an important iterative algorithm for solving monotone inclusion problems.Traditional operator splitting algorithms mainly include forward-backward splitting algorithm,Douglas-Rachford splitting algorithm and forward-backward-forward splitting algorithm.Its characteristic is that it can make full use of the attributes of each operator.Therefore,we propose an operator splitting algorithm for solving four-operator monotone inclusion containing normal cone operator.In addition,for a class of combined monotone inclusion problems with parallel sum,the existing operator splitting algorithms have some shortcomings,such as the range of parameters.For this reason,we propose two primal-dual splitting algorithms to solve such monotone inclusion problems.This article is divided into four chapters,the specific content is as follows:In the first chapter,we introduce the background of monotone inclusion problem and the research status of operator splitting algorithm.Then some symbols and definitions are given.Finally,we present the main contents of this paper.In the second chapter,we propose a forward-partial inverse-half-forward splitting(FPIHFS)algorithm for finding a zero of the sum of a maximally monotone operator,a monotone Lipschitzian operator,a cocoercive operator,and a normal cone of a closed vector subspace.The FPIHFS algorithm is derived from a combination of the partial inverse method with the forward-backward-half-forward splitting algorithm.As applications,we employ the proposed algorithm to solve several composite monotone inclusion problems,which include a finite sum of maximally monotone operators and parallel-sum of operators.In particular,we obtain a primal-dual splitting algorithm for solving a composite convex minimization problem,which has wide applications in many real problems.To verify the efficiency of the proposed algorithm,we apply it to solve the Projection on Minkowski sums of convex sets problem and the generalized Heron problem.Numerical results demonstrate the effectiveness of the proposed algorithm.In the third chapter,we propose two different primal-dual splitting algorithms for solving structured monotone inclusion containing cocoercive operator and parallel-sum of maximally monotone operators.The proposed primal-dual splitting algorithms are derived from two approaches: One is the preconditioned forward-backward splitting algorithm,and the other is the forward-backward-half-forward splitting algorithm.Both algorithms have a simple calculation framework.In particular,the single-valued operators are processed via explicit steps,while the set-valued operators are computed by their resolvents.Numerical experiments on constrained image denoising problems are presented to show the performance of the proposed algorithms.The fourth chapter summarizes the full text and gives a prospect for future work.