A note on the convergence of ADMM for linearly constrained convex optimization problems

This note serves two purposes. Firstly, we construct a counterexample to show that the statement on the convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex optimization problems in a highly influential paper by Boyd et al. (Found TrendsMach Learn 3(1):1–122, 2011) can be false if no prior condition on the existence of solutions to all the subproblems involved is assumed to hold. Secondly, we present fairly mild conditions to guarantee the existence of solutions to all the subproblems of the ADMM and provide a rigorous convergence analysis on theADMMwith a computationallymore attractive large steplength that can even exceed the practically much preferred golden ratio of (1+√5)/2. The research of the first author was supported by the China Scholarship Council while visiting the National University of Singapore and the National Natural Science Foundation of China (Grant no. 11271117). The research of the second and the third authors was supported in part by the Ministry of Education, Singapore, Academic Research Fund (Grant no. R-146-000-194-112). B Liang Chen [email protected] Defeng Sun [email protected] Kim-Chuan Toh [email protected] 1 College of Mathematics and Econometrics, Hunan University, Changsha 410082, China 2 Department of Mathematics and Risk Management Institute, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, Singapore 3 Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, Singapore