A nonsmooth optimization technique in domains of positivity

TL;DR

This paper introduces a nonsmooth proximal point method tailored for convex optimization on homogeneous domains of positivity within Hadamard manifolds, leveraging the Rham decomposition theorem for symmetric spaces.

Contribution

It develops a specialized optimization technique for a class of Hadamard manifolds, including convergence analysis and iteration bounds, expanding the toolkit for convex optimization in geometric contexts.

Findings

Proves convergence of the proposed method.

Establishes global convergence for exact and inexact versions.

Provides iteration bounds under specific assumptions.

Abstract

This paper presents a nonsmooth proximal point technique for convex optimization in a special class of Hadamard manifold called homogeneous domains of positivity. The method is based on the particularization of the Rham decomposition theorem to symmetric spaces and it is applicable to the computing of minimizers for convex functions. Homogeneous domains of positivity to be considered in this work are those of specially reducible type in a sense which is introduced and largely discussed along the paper. General aspects on the technique are shown as the convergence analysis to the inner iterations of the method, the global convergence for both versions exact and inexact and, under particular assumptions, a lower bound to the number of outer iterations of the inexact version.