On the minimal displacement vector of the Douglas-Rachford operator

G. Banjac

Operations Research Letters, vol. 49, no. 2, pp. 197-200, March 2021.
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  author = {G. Banjac},
  title = {On the minimal displacement vector of the {D}ouglas-{R}achford operator},
  journal = {Operations Research Letters},
  year = {2021},
  volume = {49},
  number = {2},
  pages = {197-200},
  url = {https://doi.org/10.1016/j.orl.2021.01.003},
  doi = {10.1016/j.orl.2021.01.003}

The Douglas-Rachford algorithm can be represented as the fixed point iteration of a firmly nonexpansive operator. When the operator has no fixed points, the algorithm's iterates diverge, but the difference between consecutive iterates converges to the so-called minimal displacement vector, which can be used to certify infeasibility of an optimization problem. In this paper, we establish new properties of the minimal displacement vector, which allow us to generalize some existing results.