An Infeasible Method with Feasibility Safeguard for Nonsmooth Composite Optimization Over Manifolds
Year: 2026
Type: preprint
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 $O(\varepsilon^{-2})$ 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.
Source: arXiv (Cornell University)
Authors: Xie, Xiyua, Li, Qia
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162427350