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In this paper, we present a method for identifying infeasible, unbounded, andpathological conic programs based on Douglas-Rachford splitting, orequivalently ADMM. When an optimization program is infeasible, unbounded, orpathological, the iterates of Douglas-Rachford splitting diverge. Somewhatsurprisingly, such divergent iterates still provide useful information, whichour method uses for identification. In addition, for strongly infeasibleproblems the method produces a separating hyperplane and informs the user onhow to minimally modify the given problem to achieve strong feasibility. As afirst-order method, the proposed algorithm relies on simple subroutines, andtherefore is simple to implement and has low per-iteration cost.

A New Use of Douglas-Rachford Splitting and ADMM for Identifying Infeasible, Unbounded, and Pathological Conic Programs