
Journal of Convex Analysis 16 (2009), No. 3, 727748 Copyright Heldermann Verlag 2009 Iterative Construction of the Resolvent of a Sum of Maximal Monotone Operators Patrick L. Combettes UPMC Université Paris 06, Lab. J.L. Lions  UMR 7598, 75005 Paris, France plc@math.jussieu.fr We propose two inexact parallel splitting algorithms for computing the resolvent of a weighted sum of maximal monotone operators in a Hilbert space and show their strong convergence. We start by establishing new results on the asymptotic behavior of the DouglasRachford splitting algorithm for the sum of two operators. These results serve as a basis for the first algorithm. The second algorithm is based on an extension of a recent Dykstralike method for computing the resolvent of the sum of two maximal monotone operators. Under standard qualification conditions, these two algorithms provide a means for computing the proximity operator of a weighted sum of lower semicontinuous convex functions. We show that a version of the second algorithm performs the same task without requiring any qualification condition. In turn, this provides a parallel splitting algorithm for qualificationfree strongly convex programming. Keywords: Dykstra's algorithm, DouglasRachford algorithm, maximal monotone operator, method of partial inverses, operator splitting, proximity operator, resolvent. MSC: 47H05; 47J25, 49M29, 65K05, 90C25 [ Fulltextpdf (191 KB)] for subscribers only. 