An Infeasible Method with Feasibility Safeguard for Nonsmooth Composite Optimization Over Manifolds
Abstract
We study nonsmooth composite optimization over embedded submanifolds defined by nonlinear equality constraints. We propose a feasibility-safeguarded inexact proximal linearized method whose iterations require neither projection nor retraction onto the manifold. A computable smooth-dual residual certifies the approximate tangential step and controls the errors in both tangency and composite linearization consistency. The step is followed by a normal correction evaluated at the displaced trial point. To coordinate the resulting optimality and feasibility errors, a nonmonotone backtracking procedure reduces the tangential stepsize faster than the normal stepsize and accepts only corrected points within a prescribed constraint-violation region. Under the stated regularity and level-boundedness assumptions, we establish a uniform backtracking bound, positive lower bounds on the accepted stepsizes, boundedness of the primal and multiplier sequences, stationarity of all cluster points, and an O(ε−2) outer iteration complexity bound. The analysis does not assume compactness of the manifold or a separate global distance-to-manifold error bound. For a specified parameter regime guaranteeing full tangential steps, a KL assumption on an auxiliary function yields full-sequence convergence. Numerical experiments on sparse principal component analysis and sparse spectral clustering illustrate the performance of the method.
Related
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BibTeX
@misc{xie2026infeasible,
title = {{An Infeasible Method with Feasibility Safeguard for Nonsmooth Composite Optimization Over Manifolds}},
author = {Xiyua Xie and Qia Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.24954},
eprint = {2605.24954},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.24954},
}
RIS
TY - UNPB
TI - An Infeasible Method with Feasibility Safeguard for Nonsmooth Composite Optimization Over Manifolds
AU - Xiyua Xie
AU - Qia Li
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2605.24954
UR - https://arxiv.org/abs/2605.24954
AB - We study nonsmooth composite optimization over embedded submanifolds defined by nonlinear equality constraints. We propose a feasibility-safeguarded inexact proximal linearized method whose iterations require neither projection nor retraction onto the manifold. A computable smooth-dual residual certifies the approximate tangential step and controls the errors in both tangency and composite linearization consistency. The step is followed by a normal correction evaluated at the displaced trial point. To coordinate the resulting optimality and feasibility errors, a nonmonotone backtracking procedure reduces the tangential stepsize faster than the normal stepsize and accepts only corrected points within a prescribed constraint-violation region. Under the stated regularity and level-boundedness assumptions, we establish a uniform backtracking bound, positive lower bounds on the accepted stepsizes, boundedness of the primal and multiplier sequences, stationarity of all cluster points, and an O(ε^−2) outer iteration complexity bound. The analysis does not assume compactness of the manifold or a separate global distance-to-manifold error bound. For a specified parameter regime guaranteeing full tangential steps, a KL assumption on an auxiliary function yields full-sequence convergence. Numerical experiments on sparse principal component analysis and sparse spectral clustering illustrate the performance of the method.
ER -