Inertial Algorithm For Solving Separable Convex Optimization Problems With Linear Equality Constraints And Its Applications

Many problems in signal and image processing,machine learning,medical image reconstruction,computer vision and network communication can be attributed to solving the optimization problem of adding two or more convex functions with linear equality constraints.For instance,the stable principal component pursuit,the latent variable Gaussian graphical model selection,the robust principal component analysis model with noisy and incomplete data,and so on.The alternating direction method of multipliers is a widely used method for solving two-block separable convex minimization problems with linear equality constraints.Considering the slow convergence of the alternating direction method of multipliers and the fact that the three-block alternating direction method of multipliers is not convergent,it is of great theoretical value and practical significance to study the acceleration method of the alternating direction method of multipliers and the three-block alternating direction method of multipliers with theoretical convergence guarantee.In this paper,we propose an inertial alternating direction method of multipliers to accelerate the convergence of the alternating direction method of multipliers.Furthermore,we propose a relaxed inertial three-block alternating minimization algorithm to solve the three-block separable convex minimization problem with linear equality constraints.In order to verify the efficiency and effectiveness of the proposed algorithms,we apply the algorithms to specific practical problems.The full paper is divided into four chapters,as follows:In Chapter 1,we first introduce the research background and current situation of two and more block separable convex minimization problems with linear equality constraints.Then we give the symbols and definitions involved in the research process of this paper and related important conclusions.Finally,the main research content of this paper is described.In Chapter 2,we mainly study the two-block separable convex minimization problems with linear equality constraints.We propose an inertial alternating direction method of multipliers for solving the two-block separable convex minimization problems with linear equality constraints.The algorithm is obtained by applying the inertial Douglas-Rachford splitting algorithm to the dual problem of the original problem.we prove the convergence of the algorithm in infinite dimension Hilbert space under the appropriate parameters conditions.Then,we conduct numerical experiments on robust principal component analysis problem and compare the proposed algorithm with other existing algorithms to demonstrate the advantage of the proposed algorithm.In Chapter 3,we study the three-block separable convex minimization problem with linear equality constraints.We design a variant three-block alternating minimization algorithm based on inertial three-operator splitting algorithm.Compared with the three-block alternating direction method of multipliers,the step one of the algorithm only minimizes the Lagrange function.As a by-product,we also get Davis and Yin's relaxation algorithm.Under mild conditions,we prove the convergence of the proposed algorithm in infinite dimensional Hilbert space.Finally,numerical experiments are carried out on stable principal component pursuit problem to verify the efficiency and effectiveness of the proposed algorithm.In Chapter 4,we summarize the whole paper and give the direction of future research.