  The alternating direction method of multipliers (ADMM) has been successfully
applied to solve structured convex optimization problems due to its superior
practical performance. The convergence properties of the 2-block ADMM have been
studied extensively in the literature. Specifically, it has been proven that
the 2-block ADMM globally converges for any penalty parameter $\gamma>0$. In
this sense, the 2-block ADMM allows the parameter to be free, i.e., there is no
need to restrict the value for the parameter when implementing this algorithm
in order to ensure convergence. However, for the 3-block ADMM, Chen \etal
\cite{Chen-admm-failure-2013} recently constructed a counter-example showing
that it can diverge if no further condition is imposed. The existing results on
studying further sufficient conditions on guaranteeing the convergence of the
3-block ADMM usually require $\gamma$ to be smaller than a certain bound, which
is usually either difficult to compute or too small to make it a practical
algorithm. In this paper, we show that the 3-block ADMM still globally
converges with any penalty parameter $\gamma>0$ if the third function $f_3$ in
the objective is smooth and strongly convex, and its condition number is in
$[1,1.0798)$, besides some other mild conditions. This requirement covers an
important class of problems to be called regularized least squares
decomposition (RLSD) in this paper.
