On some perturbation properties of nonsmooth optimization on Riemannian manifolds with applications

TL;DR

This paper develops a perturbation analysis framework for nonsmooth optimization on Riemannian manifolds, connecting variational analysis concepts to algorithmic convergence and providing practical insights for optimization methods.

Contribution

It introduces manifold variational sufficiency, relates it to classical second-order conditions, and demonstrates its implications for convergence of Riemannian optimization algorithms.

Findings

Riemannian SQP achieves local superlinear and quadratic convergence.

Riemannian Augmented Lagrangian Method attains R-linear convergence.

Numerical experiments validate theoretical convergence results.

Abstract

This paper presents a perturbation analysis framework for nonsmooth optimization on connected Riemannian manifolds to bridge the gap between the rapid development of algorithmic approaches and a robust theoretical foundation. Using tangent-space local models, we transport core notions from Euclidean variational analysis, such as strong regularity, the Aubin property, and isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping, to the manifold setting. Furthermore, we introduce the manifold (strong) variational sufficiency and show that its strong version is intrinsic, i.e., independent of the chosen retraction, and for polyhedral, second-order cone, and semidefinite programs, it coincides with the manifold strong second-order sufficient condition. These insights yield concrete algorithmic consequences. We show that the Riemannian Sequential Quadratic Programming achieves local superlinear and, under mild additional assumptions, quadratic convergence without strict complementarity, while the Riemannian Augmented Lagrangian Method attains R-linear convergence even when Lagrange multipliers are nonunique. Moreover, the proposed condition guarantees positive definiteness of the generalized Hessians associated with the augmented Lagrangian, enabling superlinear semismooth Newton steps in inner solves. Numerical experiments on robust matrix completion and compressed modes validate the theoretical predictions.