In this paper, we present a method for identifying infeasible, unbounded, and pathological conic programs based on Douglas-Rachford splitting, or equivalently ADMM. When an optimization program is infeasible, unbounded, or pathological, the iterates of Douglas-Rachford splitting diverge.Somewhat surprisingly, such divergent iterates still provide useful information, which our method uses for identification. In addition, for strongly infeasible problems the method produces a separating hyperplane and informs the user on how to minimally modify the given problem to achieve strong feasibility. Asa first-order method, the proposed algorithm relies on simple subroutines, and therefore is simple to implement and has low per-iteration cost.
Liu, Yanli, Ernest K. Ryu, and Wotao Yin. "A New Use of Douglas-Rachford Splitting and ADMM for Identifying Infeasible, Unbounded, and Pathological Conic Programs." arXiv preprint arXiv:1706.02374 (2017).