  In this paper, we develop a novel primal-dual semismooth Newton method for
solving linearly constrained multi-block convex composite optimization
problems. First, a differentiable augmented Lagrangian (AL) function is
constructed by utilizing the Moreau envelopes of the nonsmooth functions. It
enables us to derive an equivalent saddle point problem and establish the
strong AL duality under the Slater's condition. Consequently, a semismooth
system of nonlinear equations is formulated to characterize the optimality of
the original problem instead of the inclusion-form KKT conditions. We then
develop a semismooth Newton method, called ALPDSN, which uses purely
second-order steps and a nonmonotone line search based globalization strategy.
Through a connection to the inexact first-order steps when the regularization
parameter is sufficiently large, the global convergence of ALPDSN is
established. Under the regularity conditions, partial smoothness, the local
error bound, and the strict complementarity, we show that both the primal and
the dual iteration sequences possess a superlinear convergence rate and provide
concrete examples where these regularity conditions are met. Numerical results
on the image restoration with two regularization terms and the corrected tensor
nuclear norm problem are presented to demonstrate the high efficiency and
robustness of our ALPDSN.
