On the asymptotic behavior of the Douglas-Rachford and proximal-point algorithms for convex optimization

G. Banjac and J. Lygeros

Optimization Letters, vol. 15, no. 8, pp. 2719-2732, November 2021.
BibTeX  URL 

  author = {G. Banjac and J. Lygeros},
  title = {On the asymptotic behavior of the {D}ouglas-{R}achford and proximal-point algorithms for convex optimization},
  journal = {Optimization Letters},
  year = {2021},
  volume = {15},
  number = {8},
  pages = {2719-2732},
  url = {https://doi.org/10.1007/s11590-021-01706-3},
  doi = {10.1007/s11590-021-01706-3}

The authors in (Banjac et al., 2019) recently showed that the Douglas-Rachford algorithm provides certificates of infeasibility for a class of convex optimization problems. In particular, they showed that the difference between consecutive iterates generated by the algorithm converges to certificates of primal and dual strong infeasibility. Their result was shown in a finite dimensional Euclidean setting and for a particular structure of the constraint set. In this paper, we extend the result to Hilbert spaces and a general nonempty closed convex set. Moreover, we show that the proximal-point algorithm applied to the set of optimality conditions of the problem generates similar infeasibility certificates.