An Inertial Parallel and Asynchronous Fixed-Point Iteration for Convex Optimization

TL;DR

This paper introduces an inertial asynchronous parallel fixed-point iteration method for convex optimization, enabling flexible coordinate updates and proving linear convergence, with practical validation in smart grid load sharing.

Contribution

It presents the first inertial asynchronous parallel fixed-point algorithm for convex optimization with convergence guarantees and bounded update intervals.

Findings

Proven linear convergence of the proposed scheme.

Implemented in a smart grid load sharing problem.

Demonstrated superiority over non-accelerated methods.

Abstract

Two characteristics that make convex decomposition algorithms attractive are simplicity of operations and generation of parallelizable structures. In principle, these schemes require that all coordinates update at the same time, i.e., they are synchronous by construction. Introducing asynchronicity in the updates can resolve several issues that appear in the synchronous case, like load imbalances in the computations or failing communication links. However, and to the best of our knowledge, there are no instances of asynchronous versions of commonly-known algorithms combined with inertial acceleration techniques. In this work we propose an inertial asynchronous and parallel fixed-point iteration from which several new versions of existing convex optimization algorithms emanate. Departing from the norm that the frequency of the coordinates' updates should comply to some prior distribution, we propose a scheme where the only requirement is that the coordinates update within a bounded interval. We prove convergence of the sequence of iterates generated by the scheme at a linear rate. One instance of the proposed scheme is implemented to solve a distributed optimization load sharing problem in a smart grid setting and its superiority with respect to the non-accelerated version is illustrated.