Geodesic
Iterative Construction of the Resolvent of a Sum of Maximal Monotone Operators
Journal of convex analysis, Tome 16 (2009) no. 3, pp. 727-748
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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 Douglas-Rachford 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 Dykstra-like 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 qualification-free strongly convex programming.
Classification : 47H05, 47J25, 49M29, 65K05, 90C25
Mots-clés : Dykstra's algorithm, Douglas-Rachford algorithm, maximal monotone operator, method of partial inverses, operator splitting, proximity operator, resolvent 
@article{JCA_2009_16_3_JCA_2009_16_3_a9,
     author = {P. L. Combettes},
     title = {Iterative {Construction} of the {Resolvent} of a {Sum} of {Maximal} {Monotone} {Operators}},
     journal = {Journal of convex analysis},
     pages = {727--748},
     year = {2009},
     volume = {16},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2009_16_3_JCA_2009_16_3_a9/}
}
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P. L. Combettes. Iterative Construction of the Resolvent of a Sum of Maximal Monotone Operators. Journal of convex analysis, Tome 16 (2009) no. 3, pp. 727-748. http://geodesic.mathdoc.fr/item/JCA_2009_16_3_JCA_2009_16_3_a9/