Infeasibility detection in the alternating direction method of multipliers for convex optimization

G. Banjac, P. Goulart, B. Stellato and S. Boyd

Journal of Optimization Theory and Applications, vol. 183, no. 2, pp. 490-519, November 2019.
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@article{BGSetal:2019,
  author = {G. Banjac and P. Goulart and B. Stellato and S. Boyd},
  title = {Infeasibility detection in the alternating direction method of multipliers for convex optimization},
  journal = {Journal of Optimization Theory and Applications},
  year = {2019},
  volume = {183},
  number = {2},
  pages = {490-519},
  url = {https://doi.org/10.1007/s10957-019-01575-y},
  doi = {10.1007/s10957-019-01575-y}
}

The alternating direction method of multipliers is a powerful operator splitting technique for solving structured optimization problems. For convex optimization problems, it is well known that the algorithm generates iterates that converge to a solution, provided that it exists. If a solution does not exist, then the iterates diverge. Nevertheless, we show that they yield conclusive information regarding problem infeasibility for optimization problems with linear or quadratic objective functions and conic constraints, which includes quadratic, second-order cone, and semidefinite programs. In particular, we show that in the limit the iterates either satisfy a set of first-order optimality conditions or produce a certificate of either primal or dual infeasibility. Based on these results, we propose termination criteria for detecting primal and dual infeasibility.