| Douglas-Rachford algorithm is an optimization method which can be used to solve feasibility problems.For the case of nonempty closed convex constraint sets,the convergence theory of DR Algorithm as well as its different extended forms has been systematically demonstrated.However,for the application of DR Algorithm in solving non-convex feasibility problems,most of the existing studies focus on problems with particular structure,and there is no global convergence result yet.This dissertation is divided into two parts.The first part mainly studies a new algorithm for solving convex feasibility problems-generalized cyclic relaxed DR Algorithm.Employing the theory of averaged operators in Hilbert spaces,we prove that the iterative sequences generated by this algorithm converges weakly to the solution of the feasibility problem.When the constraint set is affine,the aforementioned algorithm can be applied to find a best approximation point.The second part mainly considers a nonconvex feasibility problem with the intersection of a finite set and a half space.We propose a relaxed DR Algorithm for solving such feasibility problems and prove that the auxiliary sequences generated by this algorithm enjoy global convergence.Numerical experiments show that,the generalized cyclic relaxed DR Algorithm for convex feasibility problems converges faster than that of the cyclic projection method,the classical DR Algorithm and the cyclic DR Algorithm,and the relaxed DR Algorithm for non-convex feasibility problems has better performance than that of the DR Algorithm. |
