Research On The Acceleration And Application Of Two Types Of Opera Tor Splitting Algorithms

Operator splitting algorithms are popular to solve many optimization problems in signal and image processing,machine learning and medical image reconstruction.The advantage of the operator splitting algorithms is simple and can be used to solve some nonsmooth optimization problems.However,because the parameters selection was limited,the convergence speed of the operator splitting algorithms may be slower.Therefore,it is of great theoretical significance and practical application value to explore how to speed up the operator splittin galgorithms.In this paper,we discuss the accelerated study of two-type operator splitting algorithms:forward-backward splitting algorithm and three-operator splitting algorithm.the specific contents are as follows:In the first chapter,the background of operator splitting algorithms and the research status.Then some symbols,definitions and theorems involved in this paper are given.Finally,the main research contents of this paper are expounded.In the second chapter,the forward-backward operator splitting algorithm with variable metric forward-backward Splitting algorithm is popular to solve the sum of two maximally monotone operators,one of which is cocoercive.In the infinite dimensional Hilbert space,we demonstrate the convergence of forward-backward operator splitting algorithms with over-relaxed variable distance and error.Next,under weak condition of the over-relaxed parameters,we prove the weak convergence of the algorithms we proposed.As an application,we obtain a forward-backward splitting algorithm with over-relaxed variable distance and error for solving the minimization problem of the sum of two convex functions,where one of them is differentiable with a Lipschitz continuous gradient.Furthermore,we apply the algorithms to solve variational inequalities problem,constrained convex minimization problem and split Feasibility problem.Our results improve promote other existing results.By applying to LASSO problems,numerical experimental results verify the validity and superiority of the proposed algorithm.In the third chapter,we propose a class of inertial three-operator splitting algorithm.solve the sum of three maximally monotone operators,one of which is cocoercive.The convergence of the proposed algorithm is proved by applying the inertial Krasnoselskii-Mann iteration under certain conditions on the iterative parameters in real Hilbert spaces.As applications,we develop an inertial three-operator splitting algorithm to solve the convex minimization problem of the sum of three convex functions,where one of them is differentiable with Lipschitz continuous gradient.Finally,we conduct numerical experiments on a constrained image inpainting problem with nuclear norm regularization.Numerical results demonstrate the proposed inertial three-operator splitting algorithms is faster than three-operator splitting algorithms.The fourth chapter,summarizes the full text and gives a prospect for future work.