On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators

This paper shows, by means of a new type of operator called a splitting operator, that the Douglas-Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm. Therefore, applications of Douglas-Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. The approach taken here also essentially subsumes the theory of partial inverses developed by Spingarn. We show the usefulness of the connection between Douglas-Rachford splitting and the proximal point algorithm by deriving a new, generalized alternating direction method of multipliers for convex programming. Running Heads: Operator Splitting and the Proximal Point Algorithm