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

arXiv: https://arxiv.org/abs/2103.02855 · HTML: https://mlsanity.com/paper/2103.02855/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Mar 2021.

## Built on by (most cited first, 6 of 6)

- [A Riemannian ADMM](https://mlsanity.com/paper/2211.02163/) (2022)
- [Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure](https://mlsanity.com/paper/2406.09023/) (2024)
- [A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds](https://mlsanity.com/paper/2104.04199/) (2021)
- [Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints](https://mlsanity.com/paper/2604.04130/) (2026)
- [A Support-Set Algorithm for Optimization Problems with Nonnegative and Orthogonal Constraints](https://mlsanity.com/paper/2511.03443/) (2025)
- [Optimization over the intersection of manifolds](https://mlsanity.com/paper/2605.22736/) (2026)

## Abstract (arXiv)

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.