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In this paper, we develop a novel primal-dual semismooth Newton method forsolving linearly constrained multi-block convex composite optimizationproblems. First, a differentiable augmented Lagrangian (AL) function isconstructed by utilizing the Moreau envelopes of the nonsmooth functions. Itenables us to derive an equivalent saddle point problem and establish thestrong AL duality under the Slater's condition. Consequently, a semismoothsystem of nonlinear equations is formulated to characterize the optimality ofthe original problem instead of the inclusion-form KKT conditions. We thendevelop a semismooth Newton method, called ALPDSN, which uses purelysecond-order steps and a nonmonotone line search based globalization strategy.Through a connection to the inexact first-order steps when the regularizationparameter is sufficiently large, the global convergence of ALPDSN isestablished. Under the regularity conditions, partial smoothness, the localerror bound, and the strict complementarity, we show that both the primal andthe dual iteration sequences possess a superlinear convergence rate and provideconcrete examples where these regularity conditions are met. Numerical resultson the image restoration with two regularization terms and the corrected tensornuclear norm problem are presented to demonstrate the high efficiency androbustness of our ALPDSN.

An Augmented Lagrangian Primal-Dual Semismooth Newton Method for Multi-Block Composite Optimization