  We consider three challenges in multi-block Alternating Direction Method of
Multipliers (ADMM): building convergence conditions for ADMM with any block
(variable) sequence, finding available block sequences to be fit for ADMM, and
designing useful parameter controllers for ADMM with unfixed parameters. To
address these challenges, we develop a switched control framework for studying
multi-block ADMM. First, since ADMM recursively and alternately updates the
block-variables, it is converted into a discrete-time switched dynamical
system. Second, we study exponential stability and stabilizability of the
switched system for linear convergence analysis and design of ADMM by employing
switched Lyapunov functions. Moreover, linear matrix inequalities conditions
are proposed to ensure convergence of ADMM under arbitrary sequence, to find
convergent sequences, and to design the fixed parameters. These conditions are
checked and solved by employing semidefinite programming. Numerical experiments
further verify the effectiveness of our proposed theories.

Systems and Control;Optimization and Control