Research On Alternating Direction Algorithm For Solving A Class Of Nonsmooth Convex Optimization Problems

With the development of "Internet plus",a new generation of information technology,such as the internet of things,cloud computing,large data,et al.have appeared gradually.Although this brings great convenience to people’s lives,we must take into account the time of dealing with these problems when we meet such a large amount of data information,especially for solving some of the optimization problems in real life.However,the length of the calculation depends mainly on the size of the optimization problem,the structure of problem and the convergence speed of the algorithm.Generally speaking,the traditional optimization algorithm needs to know the inverse information of the hessian matrix,which may take a lot of time and energy to solve the optimization problem with complex structure and high dimension.The alternating direction algorithm can decompose the complex large scale optimization problem into many small problems to solve,so that it has a unique advantage in dealing with such optimization problems.In this paper,we consider a class of nonsmooth unconstrained optimization problems,which is a combination of a smooth convex function and a non-smooth convex function.To solving this problem,two improved alternating direction algorithms are proposed.The specific content is as follows:Based on the framework of traditional alternating direction algorithm,a modified alternating direction algorithm is proposed to solve the total variational problem.The algorithm uses the information of the current point and the information of the first two iterations to obtain the modified initial BB step,and proposes a modified alternating direction algorithm(MADM)in combination with the nonmonotone line search technique.The convergence of the algorithm is proved theoretically,and the algorithm is applied to the total variational image reconstruction problem under the condition of small scale,noiseless and large scale and noisy.The results of the reconstructed results are evaluated from the four perspectives of run time,number of iterations,relative error and image reconstruction.Compared with the alternating direction algorithm(TVAL3),the numerical results verify the effectiveness of the algorithm.Based on the framework of accelerating the alternating linearization algorithm,an inexactalternating linearization algorithm is proposed.The algorithm uses the approximate solution to replace the exact solution of the subproblem and proves that the algorithm has a fast convergence rate.Secondly,through the punishment factor for further correction,an inexact accelerating alternating linearized with backtracking method is presented,and the complexity of the algorithm is also described in detail.Finally,the algorithm is applied to the compress sensing reconstruction problem.Comparing with accelerating iterative shrinkage/thresholding method(FISTA)and accelerating alternating linearization method(FALM)in terms of the iteration time and the number of iterations,the numerical results verify the effectiveness of the algorithm.