Research On The Acceleration And Application Of Two Types Of Primal-Dual Fixed Point Algorithms

The optimization problems about the sum of two or more convex functions have received extensive attention and research in recent years.Many optimization models in practical applications can be expressed as special cases of these convex optimization problems,including signal and image processing,machine learning,and economic management,etc.However,these optimization models are usually not smooth and involve a large number of variables,traditional optimization methods cannot be used directly or are computationally expensive.To overcome these difficulties,algorithms based on the operator splitting method and primal-dual method have become a hot research topic.In this paper,we study the acceleration research of two types of primal-dual fixed point algorithms and propose the over-relaxed primal-dual fixed point algorithm and the preprocessed primal-dual fixed point algorithm.We analyze the convergence and the convergence rate of the proposed algorithms.At the same time,we apply the proposed algorithms to solve the specific image restoration problems.The full paper is made up of four chapters,the content is as follows:In the first chapter,we introduce the background of two or more convex optimization problems and the current research status of the related algorithms.Then,we enumerate some symbols,definitions and theorems involved in this paper.Finally,we explain the main research content of this paper.In the second chapter,an over-relaxed primal-dual fixed point algorithm is proposed to solve the optimization problem of the sum of two convex functions.Compared with the primal-dual fixed point algorithm,the proposed algorithm expands the selection range of relaxation parameters.By the fixed point theory of the averaged non-expansive operator,we prove the convergence of the proposed over-relaxed algorithm and together with the ergodic convergence rate.Under some strong conditions of the objective function,we prove that the algorithm has a global linear convergence rate.To verify the validity of the proposed algorithm,we apply the proposed algorithm to solve the total variation image deblurring problem.Numerical results show that the primal-dual fixed point algorithm with the relaxed parameter larger than one(i.e.,over-relaxed)converges faster than the relaxed parameter less than one.In the third chapter,we propose a preconditioned primal-dual fixed point algorithm to solve the optimization problem of the sum of three convex functions.By defining a norm,we prove the convergence of the proposed algorithm.At the same time,we establish the connection between the proposed algorithm and some other algorithms.Further,we establish the ergodic convergence rate of the proposed preconditioned algorithm and the global linear convergence rate under some strong conditions of the objective function.Finally,we apply the proposed algorithm to solve the constraint total variation image deblurring model to verify the validity of the proposed preconditioned algorithm.Numerical results show that the proposed preconditioned primal-dual fixed point algorithm outperforms other state-of-the-art primal-dual algorithms in terms of the number of iterations.The fourth chapter summarizes the full paper and gives the direction for future research work.