  In this paper, we show that Chen-Teboulle's proximal-based decomposition
method can be interpreted as a proximal augmented Lagrangian method. More
precisely, it coincides with a linearized augmented Lagrangian method. We then
proposed three generalized methods based on this interpretation. By invoking
recent work (He et al., IMA J. Numer. Anal., 32 (2020), pp. 227--245), we show
that the step size condition of Chen-Teboulle's method can be relaxed without
adding any further assumptions. Our analysis offers a new insight into this
proximal-based decomposition method.
