Some Theoretical Research Of The Generalized Alternating Direction Method Of Multipliers

In this paper,we further study the theoretical aspects of the generalized Peaceman-Rachford splitting method for solving a convex minimization model with two-block sepa-rable structures.And we also develop a new linearized version of generalized alternating direction method of multipliers(L-GADMM)for the linearly constrained separable convex programming whose objective functions are the sum of three convex functions without coupled variables.Firstly,the global convergence of the generalized Peaceman-Rachford splitting method is well know,while research on its convergence rate is still in its infancy.In this thesis,we first establish the worst-case(?)(1/t)convergence rate for the proposed GPRSM in both the ergodic and nonergodic senses,then we give some numerical results to demonstrate the convergence rate of the algorithm.And then,we extend the L-GADMM directly to the case of a three-block convex minimization problem where its objective function is the sum of three separable convex functions.However,the convergence of this extension can't be ensured under no extra assumptions.We give a sufficient condition to ensure the convergence of the L-GADMM for three-block separable convex minimization problem.Theoretically,we establish the worst-case(?)(1/t)convergence rate for the proposed L-GADMM in both ergodic and nonergodic senses under the sufficient condition.Moreover,we also show an example to prove its divergence of the proposed L-GADMM if the sufficient condition is lost and give some numerical results.