Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions

Full Length Paper Series B

DOI: 10.1007/s10107-016-1044-0

Cite this article as:
Combettes, P.L. & Eckstein, J. Math. Program. (2016). doi:10.1007/s10107-016-1044-0


We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. The principal innovation in these algorithms is that they are block-iterative in the sense that, at each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in established methods. Flexible strategies are used to select the blocks of operators activated at each iteration. In addition, we allow lags in operator processing, permitting asynchronous implementation. The decomposition phase of each iteration of our methods is to generate points in the graphs of the selected monotone operators, in order to construct a half-space containing the Kuhn–Tucker set associated with the system. The coordination phase of each iteration involves a projection onto this half-space. We present two related methods: the first method provides weakly convergent primal and dual sequences under general conditions, while the second is a variant in which strong convergence is guaranteed without additional assumptions. Neither algorithm requires prior knowledge of bounds on the linear operators involved or the inversion of linear operators. Our algorithmic framework unifies and significantly extends the approaches taken in earlier work on primal-dual projective splitting methods.


Asynchronous algorithm Block-iterative algorithm Duality Monotone inclusion Monotone operator Primal-dual algorithm Splitting algorithm 

Mathematics Subject Classification

47H05 49M27 65K05 90C25 

Funding information

Funder NameGrant NumberFunding Note
  • 2013MesureHD
CNRS Imag’in
National Science Foundation
  • CCF-1115638

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis Lions – UMR CNRS 7598UPMC Université Paris 06ParisFrance
  2. 2.Department of Managemement Science and Information Systems and RUTCORRutgers UniversityPiscatawayUSA

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