An analysis of the superiorization method via the principle of concentration of measure

TL;DR

This paper analyzes the superiorization method, which adapts feasibility-seeking algorithms to reduce target function values without full minimization, using the principle of concentration of measure to address theoretical guarantees.

Contribution

It introduces an analysis framework based on the concentration of measure principle to better understand the theoretical guarantees of the superiorization method.

Findings

Provides a new theoretical perspective on superiorization

Suggests conditions under which target function reduction accumulates

Enhances understanding of the method's reliability

Abstract

The superiorization methodology is intended to work with input data of constrained minimization problems, i.e., a target function and a constraints set. However, it is based on an antipodal way of thinking to the thinking that leads constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibility-seeking algorithms to reduce (not necessarily minimize) target function values. This is done while retaining the feasibility-seeking nature of the algorithm and without paying a high computational price. A guarantee that the local target function reduction steps properly accumulate to a global target function value reduction is still missing in spite of an ever-growing body of publications that supply evidence of the success of the superiorization method in various problems. We propose an analysis based on the principle of concentration of measure that attempts to alleviate the guarantee question of the superiorization method.