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

Zhanwang Deng, Kangkang Deng, Jiang Hu, Zaiwen Wen

Abstract: 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 partial smoothness, the local error bound, and the strict complementarity conditions, 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 demonstrate the high efficiency and robustness of the proposed ALPDSN.