On the proximal point algorithm for strongly quasiconvex functions in Hadamard spaces

TL;DR

This paper proves the convergence of the proximal point algorithm for strongly quasiconvex functions in Hadamard spaces, providing new elementary proofs and establishing fast convergence rates, including linear rates, even in Euclidean spaces.

Contribution

It introduces a simple, effective convergence proof for the proximal point algorithm in Hadamard spaces, with novel linear convergence rate results.

Findings

Proved convergence of the proximal point algorithm in Hadamard spaces.

Established fast convergence rates, including linear rates.

Provided elementary proofs and new properties of proximal operators in nonlinear spaces.

Abstract

We prove the convergence of the proximal point algorithm for finding the unique minimizer of a strongly quasiconvex function in general nonlinear Hadamard spaces, generalizing a recent result due to F. Lara. Our argument is rather elementary and brief and relies only on a few properties of strongly quasiconvex functions and their proximal operators which are established here for the first time over these nonlinear spaces. In particular, our convergence proof is fully effective and actually yields fast (ranging up to linear) rates of convergence for the iterates towards the solution and for the function values towards the minimum. These rates are novel even in the context of Euclidean spaces.