Perturbation Resilience and Superiorization of Iterative Algorithms

TL;DR

This paper introduces a methodology to enhance iterative algorithms by making them 'superiorized' for better optimization performance while maintaining efficiency, especially when the original algorithms are perturbation resilient.

Contribution

It presents a novel superiorization technique that automatically modifies existing perturbation resilient algorithms to improve their optimization capabilities without increasing computational resources.

Findings

Superiority is achieved through perturbations guiding algorithms towards optimal solutions.

Various projection algorithms are shown to be perturbation resilient.

Application to image reconstruction demonstrates practical effectiveness.

Abstract

Iterative algorithms aimed at solving some problems are discussed. For certain problems, such as finding a common point in the intersection of a finite number of convex sets, there often exist iterative algorithms that impose very little demand on computer resources. For other problems, such as finding that point in the intersection at which the value of a given function is optimal, algorithms tend to need more computer memory and longer execution time. A methodology is presented whose aim is to produce automatically for an iterative algorithm of the first kind a "superiorized version" of it that retains its computational efficiency but nevertheless goes a long way towards solving an optimization problem. This is possible to do if the original algorithm is "perturbation resilient," which is shown to be the case for various projection algorithms for solving the consistent convex feasibility problem. The superiorized versions of such algorithms use perturbations that drive the process in the direction of the optimizer of the given function. After presenting these intuitive ideas in a precise mathematical form, they are illustrated in image reconstruction from projections for two different projection algorithms superiorized for the function whose value is the total variation of the image.