Random Block Coordinate Descent Methods for Linearly Constrained Optimization over Networks

  • Ion Necoara
  • Yurii Nesterov
  • François Glineur


In this paper we develop random block coordinate descent methods for minimizing large-scale linearly constrained convex problems over networks. Since coupled constraints appear in the problem, we devise an algorithm that updates in parallel at each iteration at least two random components of the solution, chosen according to a given probability distribution. Those computations can be performed in a distributed fashion according to the structure of the network. Complexity per iteration of the proposed methods is usually cheaper than that of the full gradient method when the number of nodes in the network is much larger than the number of updated components. On smooth convex problems, we prove that these methods exhibit a sublinear worst-case convergence rate in the expected value of the objective function. Moreover, this convergence rate depends linearly on the number of components to be updated. On smooth strongly convex problems we prove that our methods converge linearly. We also focus on how to choose the probabilities to make our randomized algorithms converge as fast as possible, which leads us to solving a sparse semidefinite program. We then describe several applications that fit in our framework, in particular the convex feasibility problem. Finally, numerical experiments illustrate the behaviour of our methods, showing in particular that updating more than two components in parallel accelerates the method.


Convex optimization over networks Linear coupled constraints Random coordinate descent Distributed computations Convergence analysis 

Mathematics Subject Classification

90C06 90C25 90C35 



This research received funding from UEFISCDI Romania, PNII-RU-TE, project MoCOBiDS, no. 176/01.10.2015. It presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, and of the Concerted Research Action (ARC) programme supported by the Federation Wallonia-Brussels (contract ARC 14/19-060). Support from two WBI-Romanian Academy grants is also acknowledged. Scientific responsibility rests with the authors.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Automatic Control and Systems Engineering DepartmentUniversity Politehnica BucharestBucharestRomania
  2. 2.Center for Operations Research and Econometrics, ICTEAM InstituteUniversite Catholique de LouvainLouvain-la-NeuveBelgium

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