The Research Of Alternating Direction Methods And Their Modification

Alternating direction method is a decomposition method to obtain the solution ofthe original problem via alternately solving a series of sub-problems. It is one of themost popular method for solving monotone varaiational inequalities with separablestructure and linearly constrained structured convex optimization.At first, to solve monotone varaiational inequalities with separable structure, Yuan[1]pretents a PADM-based descent-type method in2011. In Chapter2, by choosing adifferent descent direction and identifing the step size along the descent direction, a newproximal alternating direction method pretents. Moreover, we proved our method isbetter than Yuan’s mtheod in theoretical senses.Secondly, in Chapter3, for solving linearly constrained structured convexoptimization, we gives an improved linearized proximal alternating direction method.The new method designs the descent direction by using the iterates generated bylinearized proximal alternating direction method and identifies the step size along thedescent direction by maximizing the merit function. Convergence of the new method isproved under appropriate assumptions. Numerical result demonstrates that the newmethod is efficient.