On some perturbation properties of nonsmooth optimization on Riemannian manifolds with applications
Yuexin Zhou, Chao Ding, Yangjing Zhang
TL;DR
The paper develops a retraction-based perturbation framework for nonsmooth optimization on connected Riemannian manifolds, enabling tangent-space surrogate analysis of KKT regularity and perturbation behavior. It defines manifold counterparts to Euclidean notions such as RCQ, SRCQ, SOSC, and SSOSC and introduces manifold (strong) variational sufficiency, establishing deep links to augmented Lagrangian convexity and augmented tilt stability. These theoretical tools yield concrete convergence guarantees: RSQP achieves local superlinear (quadratic under certain settings) convergence under robust isolated calmness and M-SRCQ+M-SOSC, while RALM attains outer R-linear convergence with inner semismooth Newton steps converging superlinear under manifold variational sufficiency. Numerical experiments on robust matrix completion and compressed modes corroborate the theory and demonstrate the practical viability of the proposed perturbation and convergence frameworks for manifold nonsmooth optimization.
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.