Attouch–Théra duality revisited: Paramonotonicity and operator splitting

The problem of finding the zeros of the sum of two maximally monotone operators is of fundamental importance in optimization and variational analysis. In this paper, we systematically study Attouch–Théra duality for this problem. We provide new results related to Passty’s parallel sum, to Eckstein and Svaiter’s extended solution set, and to Combettes’ fixed point description of the set of primal solutions. Furthermore, paramonotonicity is revealed to be a key property because it allows for the recovery of all primal solutions given just one arbitrary dual solution. As an application, we generalize the best approximation results by Bauschke, Combettes and Luke [H.H. Bauschke, P.L. Combettes, D.R. Luke, A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space, Journal of Approximation Theory 141 (2006) 63–69] from normal cone operators to paramonotone operators. Our results are illustrated through numerous examples.