A Busemann hybrid projection-proximal point algorithm for optimization problems on Hadamard manifolds

TL;DR

This paper introduces a new geometric optimization algorithm on Hadamard manifolds using Busemann functions, achieving convergence and complexity results without tangent space linear solves.

Contribution

It proposes the Busemann hybrid projection proximal point algorithm, a novel method replacing hyperplanes with horospheres, and analyzes its convergence with inexact subgradient evaluations.

Findings

Global convergence under controlled inexactness

Sublinear complexity rate proportional to inverse square root of iterations

Exact variant matches classical Riemannian proximal point algorithm

Abstract

We study optimization problems on Hadamard manifolds, motivated by recent advances in geometric approaches to optimization on curved spaces, particularly those involving the structure of Busemann functions. We introduce a projection based variant of the proximal point algorithm, termed the \emph{Busemann hybrid projection proximal point algorithm}, which replaces Euclidean hyperplanes with horospheres defined via convex Busemann functions. The algorithm performs projections in closed form using the gradients of these functions, resulting in a geometrically intrinsic scheme that requires no tangent space linear solves. We allow for inexact subgradient evaluations and prove global convergence under controlled inexactness, with a relative error level strictly below one. We establish a Fej\'er type descent and sublinear complexity with a rate proportional to the inverse square root of the iteration count, and show that the exact variant coincides with the classical Riemannian proximal point algorithm. The framework clarifies the role of Busemann based subdifferentials in optimization on spaces of nonpositive curvature.