Second order semi-smooth Proximal Newton methods in Hilbert spaces

TL;DR

This paper introduces a globalized Proximal Newton method for complex minimization problems in Hilbert spaces, leveraging second order semi-smoothness to ensure convergence and acceleration under relaxed assumptions.

Contribution

It develops a novel Proximal Newton approach using second order semi-smoothness, broadening applicability to non-convex and less differentiable functionals in Hilbert spaces.

Findings

Method converges globally and locally accelerates.

Applicable to non-convex, non-smooth problems in function spaces.

Demonstrated on a toy model problem.

Abstract

We develop a globalized Proximal Newton method for composite and possibly non-convex minimization problems in Hilbert spaces. Additionally, we impose less restrictive assumptions on the composite objective functional considering differentiability and convexity than in existing theory. As far as differentiability of the smooth part of the objective function is concerned, we introduce the notion of second order semi-smoothness and discuss why it constitutes an adequate framework for our Proximal Newton method. However, both global convergence as well as local acceleration still pertain to hold in our scenario. Eventually, the convergence properties of our algorithm are displayed by solving a toy model problem in function space.