Ball-proximal point method on a Hadamard Manifolds

TL;DR

This paper introduces a Riemannian ball-proximal point method for minimizing geodesically convex functions on Hadamard manifolds, with convergence guarantees and complexity bounds.

Contribution

It extends Euclidean ball-proximal ideas to Hadamard manifolds, establishing existence, uniqueness, and convergence properties of the proposed method.

Findings

The method guarantees a strict decrease in distance to the solution set.

Finite termination occurs with constant radii.

Linear decay of objective values up to the solution.

Abstract

We consider the problem of minimizing a proper, lower semicontinuous, geodesically convex function on a Hadamard manifold. Building on ball-proximal (broximal) ideas in the Euclidean setting, viewed as an abstract proximal-type algorithm, we propose and analyze a Riemannian ball-proximal point method (RB-PPM) whose basic step consists of minimizing the objective function over a metric ball centred at the current iterate. We first introduce the Riemannian broximal map, prove existence and uniqueness of broximal points on Hadamard manifolds, and derive a KKT-type characterization involving a scalar parameter and the Riemannian subdifferential. We then show that RB-PPM enjoys a strict decrease of the squared distance to the solution set whenever the current ball does not contain a minimizer. This leads to quasi-Fej\'er monotonicity, finite termination for constant radii, and a product-form linear decay of the objective values up to the hitting time of the solution set. We also obtain nonasymptotic complexity bounds for the norms of suitable subgradients and for the function values, including a linear rate in the number of iterations under constant radii. Finally, we establish an asymptotic dichotomy, if the sum of the radii diverges, then the objective values converge to the optimal value, and, when the solution set is nonempty, the entire sequence of iterates converges to a minimizer. The resulting scheme provides a geometry-aware, ball-based analog of classical Riemannian proximal point methods.