Tight Global Linear Convergence Rate Bounds for Operator Splitting Methods

G. Banjac and P. J. Goulart

IEEE Transactions on Automatic Control, vol. 63, no. 12, pp. 4126-4139, December 2018.
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  author = {G. Banjac and P. J. Goulart},
  title = {Tight Global Linear Convergence Rate Bounds for Operator Splitting Methods},
  journal = {IEEE Transactions on Automatic Control},
  year = {2018},
  volume = {63},
  number = {12},
  pages = {4126-4139},
  url = {https://doi.org/10.1109/TAC.2018.2808442},
  doi = {10.1109/TAC.2018.2808442}

In this paper we establish necessary and sufficient conditions for linear convergence of operator splitting methods for a general class of convex optimization problems where the associated fixed-point operator is averaged. We also provide a tight bound on the achievable convergence rate. Most existing results establishing linear convergence in such methods require restrictive assumptions regarding strong convexity and smoothness of the constituent functions in the optimization problem. However, there are several examples in the literature showing that linear convergence is possible even when these properties do not hold. We provide a unifying analysis method for establishing linear convergence based on linear regularity and show that many existing results are special cases of our approach. Moreover, we propose a novel linearly convergent splitting method for linear programming.