The Degenerate Variable Metric Proximal Point Algorithm and Adaptive Stepsizes for Primal-Dual Douglas-Rachford

TL;DR

This paper introduces a novel degenerate variable metric proximal point algorithm with adaptive stepsizes, combining preconditioning and primal-dual methods, and proves its weak convergence to improve optimization efficiency.

Contribution

It develops a new algorithm that integrates variable preconditioning with primal-dual Douglas-Rachford, providing convergence proofs and heuristics for faster convergence.

Findings

Proves weak convergence of the proposed algorithm.

Derives a primal-dual Douglas-Rachford variant with varying preconditioners.

Provides heuristics for selecting preconditioners to enhance convergence speed.

Abstract

In this paper the degenerate preconditioned proximal point algorithm will be combined with the idea of varying preconditioners leading to the degenerate variable metric proximal point algorithm. The weak convergence of the resulting iteration will be proven. From the perspective of the degenerate variable metric proximal point algorithm, a version of the primal-dual Douglas-Rachford method with varying preconditioners will be derived and a proof of its weak convergence which is based on the previous results for the proximal point algorithm, is provided, too. After that, we derive a heuristic on how to choose those varying preconditioners in order to increase the convergence speed of the method.