The Riemannian Convex Bundle Method

2024/02/21 by Ronny Bergmann, Roland Herzog, Bergmann, Ronny +3 · 3 citations
Computer Science · Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Optimization and Control (math.OC) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.13670

openalex publication_date 2024/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the convex bundle method to solve convex, non-smooth optimization problems on Riemannian manifolds of bounded sectional curvature. Each step of our method is based on a model that involves the convex hull of previously collected subgradients, parallelly transported into the current serious iterate. This approach generalizes the dual form of classical bundle subproblems in Euclidean space. We prove that, under mild conditions, the convex bundle method converges to a minimizer. Several numerical examples implemented using Manopt.jl illustrate the performance of the proposed method and compare it to the subgradient method, the cyclic proximal point algorithm, as well as the proximal bundle method.

Cited by

• Iso-Riemannian Optimization on Learned Data Manifolds
• Procrustes Problems on Random Matrices
• The Intrinsic Riemannian Proximal Gradient Method for Nonconvex Optimization

Related

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• An accelerated proximal bundle method for convex optimization
• Proximal bundle algorithms for nonsmooth convex optimization via fast gradient smooth methods
• Iteration-Complexity of the Subgradient Method on Riemannian Manifolds with Lower Bounded Curvature
• Optimal Convergence Rates for the Proximal Bundle Method