A Note on the Convergence of ADMM for Linearly Constrained Convex Optimization Problems

摘要

This note serves two purposes. Firstly, we construct a counterexample to showthat the statement on the convergence of the alternating direction method ofmultipliers (ADMM) for solving linearly constrained convex optimizationproblems in a highly influential paper by Boyd et al. [Found. Trends Mach.Learn. 3(1) 1-122 (2011)] can be false if no prior condition on the existenceof solutions to all the subproblems involved is assumed to hold. Secondly, wepresent fairly mild conditions to guarantee the existence of solutions to allthe subproblems and provide a rigorous convergence analysis on the ADMM, undera more general and useful semi-proximal ADMM (sPADMM) setting considered byFazel et al. [SIAM J. Matrix Anal. Appl. 34(3) 946-977 (2013)], with acomputationally more attractive large step-length that can even exceed thepractically much preferred golden ratio of $(1+\sqrt{5})/2$.