Dynamical Alternating Direction Method Of Multipliers For Two Kinds Of Block Optimization Problems

Block optimization problems are widely used in practical applications,such as compressed sensing and image processing.Based on the study of first-order dynamical systems with convex optimization problems,it is found that studying optimization algorithms from the perspective of differential equations not only helps to understand the essence of algorithms,but also provides an effective tool for the analysis of convergence of algorithms.This dissertation mainly explores the theory of dynamical alternating direction method of multiplier to solve the linear constrained separable convex optimization problems.Firstly,the general two-block convex optimization problem with linear constraints is considered.A dynamical system in which the discrete case is the classical alternating direction method of multiplier is constructed.By using the Cauchy-Lipschitz-Picard theorem,the existence and uniqueness of the solution trajectory of the system are proved;Then,by constructing the energy function,using the Lyapunov stability theory,it is proved that the solution trajectory of the system weakly converges to the saddle point of the Lagrange function of the problem;Furthermore,when the coefficient matrices of the problem constraints are identity matrices,the convergence rate of the solution trajectory in the sense of ergodic is proved.In addition,under the condition of the lack of strong convexity of the objective function,when one of the coefficient matrices of the problem constraints is the identity matrix,the convergence analysis of the solution trajectory of the system is given.Finally,the effectiveness of the dynamical system is verified by numerical experiments.Secondly,the linear constrained two-block convex optimization problem with differentiable terms is considered.Combined with the idea of linearization,a dynamical system in which the discrete case is the linearized alternating direction method of multiplier is constructed.Under appropriate conditions,the existence and uniqueness,convergence and convergence rate of solution trajectories of the system are proved.