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 $O({\epsilon }^{-2})$ 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.