A necessary condition for the guarantee of the superiorization method

TL;DR

This paper identifies a necessary condition for the superiorization method to guarantee improved objective function values, highlighting a negative condition that must be avoided for successful application.

Contribution

It presents a new necessary condition under which the superiorization method fails to produce better outcomes, guiding future guarantees and practical use.

Findings

A negative gradient descent step can prevent superior outcomes.

The identified condition is necessary for guaranteeing the method's success.

Practitioners can avoid this condition to improve real-world application success.

Abstract

We study a method that involves principally convex feasibility-seeking and makes secondary efforts of objective function value reduction. This is the well-known superiorization method (SM), where the iterates of an asymptotically convergent iterative feasibility-seeking algorithm are perturbed by objective function nonascent steps. We investigate the question under what conditions a sequence generated by an SM algorithm asymptotically converges to a feasible point whose objective function value is superior (meaning smaller or equal) to that of a feasible point reached by the corresponding unperturbed one (i.e., the exactly same feasibility-seeking algorithm that the SM algorithm employs.) This question is yet only partially answered in the literature. We present a condition under which an SM algorithm that uses negative gradient descent steps in its perturbations fails to yield such a superior outcome. The significance of the discovery of this negative condition is that it necessitates that the inverse of this condition will have to be assumed to hold in any future guarantee result for the SM. The condition is important for practitioners who use the SM because it is avoidable in experimental work with the SM, thus increasing the success rate of the method in real-world applications.