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<p>We generalize tilt stability, a fundamental concept in perturbation analysis of optimization problems in Euclidean spaces, to the setting of Riemannian manifolds. We prove the equivalence of the following conditions: Riemannian tilt stability, Riemannian variational strong convexity, Riemannian uniform quadratic growth, local maximal strong monotonicity of Riemannian subdifferential, strong metric regularity of Riemannian subdifferential (in addition to local minimality), and positive definiteness of generalized Riemannian Hessian. For Riemannian nonlinear programming, we provide a characterization of Riemannian tilt stability under a weak constraint qualification. Leveraging these results, we propose a generalized Riemannian Newton method and establish its superlinear convergence under Riemannian tilt stability.</p>

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Tilt Stability on Riemannian Manifolds with Application to Convergence Analysis of Generalized Riemannian Newton Method

  • 期刊论文

  • 发表日期：

    2026年6月19日

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  • Zijian Shi,
  • Miantao Chao

摘要

We generalize tilt stability, a fundamental concept in perturbation analysis of optimization problems in Euclidean spaces, to the setting of Riemannian manifolds. We prove the equivalence of the following conditions: Riemannian tilt stability, Riemannian variational strong convexity, Riemannian uniform quadratic growth, local maximal strong monotonicity of Riemannian subdifferential, strong metric regularity of Riemannian subdifferential (in addition to local minimality), and positive definiteness of generalized Riemannian Hessian. For Riemannian nonlinear programming, we provide a characterization of Riemannian tilt stability under a weak constraint qualification. Leveraging these results, we propose a generalized Riemannian Newton method and establish its superlinear convergence under Riemannian tilt stability.