Infeasibility Detection in the Alternating Direction Method of Multipliers for Convex Optimization

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

Working draft, March 2018.

The alternating direction method of multipliers (ADMM) is a powerful operator splitting technique for solving structured optimization problems. For convex optimization problems, it is well-known that the iterates generated by ADMM converge to a solution provided that it exists. If a solution does not exist, then the ADMM iterates do not converge. Nevertheless, we show that the ADMM iterates yield conclusive information regarding problem infeasibility for a wide class of convex optimization problems including both quadratic and conic programs. In particular, we show that in the limit the ADMM 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 in ADMM.