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

TL;DR

This paper investigates the robust isolated calmness of KKT solution mappings in nonsmooth optimization on Riemannian manifolds, establishing conditions for stability and convergence of algorithms with applications to specific geometric problems.

Contribution

It introduces manifold versions of constraint qualifications and second order conditions, linking them to solution stability and convergence in Riemannian nonsmooth optimization.

Findings

Robust isolated calmness is equivalent to M-SRCQ and M-SOSC conditions.

Under these conditions, the Riemannian augmented Lagrangian method converges linearly.

Validated conditions and convergence rate on sphere and fixed rank matrix problems.

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.