Retraction-Free Nonsmooth Optimization on the Stiefel Manifold: Subgradient and Proximal Landing Methods
Abstract
We develop three retraction-free Landing methods for nonsmooth composite optimization over the Stiefel manifold, combining an (approximately) tangential direction with an infeasibility-reducing normal direction. The subgradient Landing method, with normal stepsizes parameterized by α∈[0,1], has best-iterate complexity bounds 𝒪(ε−4) for surrogate stationarity and 𝒪(ε−2/(2−α)) for feasibility. For α=0, it converges locally R-linearly under local weak sharpness and mild assumptions. The proximal gradient Landing method computes a tangential direction by solving a proximal subproblem and combines it with the same normal direction. Moreover, the inexact proximal gradient Landing method allows the proximal subproblem to be solved approximately and applies an explicit correction to control violations of the tangency condition. Both proximal methods have outer-iteration complexity bounds 𝒪(ε−2) for stationarity and 𝒪(ε−1) for feasibility, and every accumulation point is first-order stationary. Under an additional asymptotic Bregman-error condition and the Kurdyka-Lojasiewicz property of a suitable auxiliary function, the primal sequence of the inexact method converges to a first-order stationary point. Numerical experiments illustrate the effectiveness of all three methods. On the tested sparse principal component analysis instances, the inexact method has favorable running times, and both proximal methods maintain small feasibility residuals.
Related
- Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems
- Linear convergence rate analysis of Proximal-Tracking for nonsmooth distributed optimization
- Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization
- A Proximal Gradient Method for Nonconvex Nonsmooth Optimization with Applications
- Weak Convexity and Proximal Bundle Methods for Nonsmooth Policy Optimization in Robust Control
BibTeX
@misc{cui2026retractionfree,
title = {{Retraction-Free Nonsmooth Optimization on the Stiefel Manifold: Subgradient and Proximal Landing Methods}},
author = {Zilong Cui and Tongxin Hu and Bin Gao and Ran Gu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30617},
eprint = {2610.30617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30617},
}
RIS
TY - UNPB
TI - Retraction-Free Nonsmooth Optimization on the Stiefel Manifold: Subgradient and Proximal Landing Methods
AU - Zilong Cui
AU - Tongxin Hu
AU - Bin Gao
AU - Ran Gu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30617
UR - https://arxiv.org/abs/2610.30617
AB - We develop three retraction-free Landing methods for nonsmooth composite optimization over the Stiefel manifold, combining an (approximately) tangential direction with an infeasibility-reducing normal direction. The subgradient Landing method, with normal stepsizes parameterized by α∈[0,1], has best-iterate complexity bounds 𝒪(ε^−4) for surrogate stationarity and 𝒪(ε^−2/(2−α)) for feasibility. For α=0, it converges locally R-linearly under local weak sharpness and mild assumptions. The proximal gradient Landing method computes a tangential direction by solving a proximal subproblem and combines it with the same normal direction. Moreover, the inexact proximal gradient Landing method allows the proximal subproblem to be solved approximately and applies an explicit correction to control violations of the tangency condition. Both proximal methods have outer-iteration complexity bounds 𝒪(ε^−2) for stationarity and 𝒪(ε^−1) for feasibility, and every accumulation point is first-order stationary. Under an additional asymptotic Bregman-error condition and the Kurdyka-Lojasiewicz property of a suitable auxiliary function, the primal sequence of the inexact method converges to a first-order stationary point. Numerical experiments illustrate the effectiveness of all three methods. On the tested sparse principal component analysis instances, the inexact method has favorable running times, and both proximal methods maintain small feasibility residuals.
ER -