A Semismooth Newton based Augmented Lagrangian Method for Nonsmooth Optimization on Matrix Manifolds

TL;DR

This paper introduces a novel augmented Lagrangian method combined with a semismooth Newton approach for nonsmooth optimization problems on matrix manifolds, demonstrating convergence and efficiency through theoretical analysis and numerical experiments.

Contribution

It develops a new semismooth Newton based augmented Lagrangian method for nonsmooth manifold optimization, with proven convergence and practical effectiveness.

Findings

Method converges to stationary points under certain conditions

Achieves local superlinear convergence with the semismooth Newton method

Numerical experiments show advantages in compressed modes and sparse PCA

Abstract

This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and nonlinear constraints. Under the constant positive linear dependence condition on manifolds, we show that the proposed method converges to a stationary point of the nonsmooth manifold optimization problem. Moreover, we propose a globalized semismooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semismooth Newton method is also established under some suitable conditions. We also prove that the semismoothness on submanifolds can be inherited from that in the ambient manifold. Finally, numerical experiments on compressed modes and (constrained) sparse principal component analysis illustrate the advantages of the proposed method.