Proximal Decomposition on the Graph of a Maximal Monotone Operator

We present an algorithm to solve: Find ( x,y ) ∈ A × A^ such that y ∈ Tx, where A is a subspace and T is a maximal monotone operator. The algorithm is based on the proximal decomposition on the graph of a monotone operator and we show how to recover Spingarn’s decomposition method. We give a proof of convergence that does not use the concept of partial inverse and show how to choose a scaling factor to accelerate the convergence in the strongly monotone case. Numerical results performed on quadratic problems confirm the robust behaviour of the algorithm.