The problem frequently arising in engineering mathematics is finding a point in the intersection of two or more closed sets. The Douglas-Rachford algorithm and the method of alternating projection are the two frequently used projection algorithms to solve the feasibility problems. It is noticed that both the algorithms are regularly utilized to solve feasibility problems involving one and more sets that are nonconvex but without theoretical justification. Motivated by this, we considered the convergence of the algorithms by taking one of the sets as nonconvex.
The Douglas-Rachford splitting algorithm is successfully implemented in the convex feasibility problems. When one or more set is nonconvex, the algorithm is used to solve different types of feasibility problems, but the algorithm's convergence is still not thoroughly investigated. We studied the behavior of the Douglas-Rachford method in both consistent and inconsistent settings. Interestingly, the Douglas-Rachford operator is not a nonexpansive map which is an essential requirement for studying the convergence of the DRA. Also, we have shown that, in the consistent case, the Douglas-Rachford sequence converges, and the corresponding shadow sequence converges to the intersection of sets. In the inconsistent case, we have proved that the Douglas-Rachford sequence diverges, and the corresponding shadow sequence has a weak cluster point.
The method of alternating projection is the simplest projection algorithm to find a point in the intersection of two or more nonempty sets. It involves finding orthogonal projection alternately onto two or more closed sets in a Hilbert space. Norm convergence of the method is established in a convex setting for periodic or quasi-periodic projection provided the intersection of the sets is nonempty. We found the region for global convergence in a nonconvex setting. Few examples and figures are also provided to illustrate and validate the results