On the robust isolated calmness of a class of nonsmooth optimizations on Riemannian manifolds and its applications

TL;DR

This work studies perturbation analysis and convergence for nonsmooth optimization problems on Riemannian manifolds. It considers problems of the form $\min f(x)+\theta(g_1(x))$ subject to $g_2(x) \in \mathcal{Q}$ and $x \in \mathcal{M}$, with $\theta$ convex and $f,g_1,g_2$ smooth. It introduces manifold analogues M-RCQ, M-SRCQ, M-SOSC, and proves that robust isolated calmness of the KKT solution mapping is equivalent to M-SRCQ and M-SOSC; under these conditions the Riemannian Augmented Lagrangian Method achieves local linear convergence. The approach leverages a perturbation framework via normal coordinate charts to relate the manifold conditions to Euclidean RCQ/SOSC and validates the theory with sphere and fixed-rank matrix experiments. The results extend perturbation analysis and linear convergence of ALM from Euclidean to Riemannian settings and provide a foundation for robust and efficient algorithms on manifolds.

Abstract

This paper studies the robust isolated calmness property of the KKT solution mapping of a class of nonsmooth optimization problems on Riemannian manifolds. The manifold versions of the Robinson constraint qualification, the strict Robinson constraint qualification, and the second order conditions are defined and discussed. We show that the robust isolated calmness of the KKT solution mapping is equivalent to satisfying the M-SRCQ and M-SOSC conditions. Furthermore, under the above two conditions, we show that the Riemannian augmented Lagrangian method achieves a local linear convergence rate. Finally, we verify the proposed conditions and demonstrate the convergence rate on two minimization problems over the sphere and the manifold of fixed rank matrices.