# A necessary condition for the guarantee of the superiorization method

## TL;DR

This paper identifies a condition under which a modified optimization method fails, helping improve its real-world effectiveness.

## Contribution

The paper introduces a 'negative condition' that must be avoided for the superiorization method to succeed.

## Key findings

- An SM algorithm using negative gradient descent steps fails under a specific condition.
- The identified condition is avoidable in practice, increasing the method's success rate.
- Future guarantees for the SM must assume the inverse of this condition.

## 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.