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A necessary condition for the guarantee of the superiorization method

Kay Barshad, Yair Censor, Walaa Moursi, Tyler Weames, Henry Wolkowicz

TL;DR

This paper identifies a negative condition under which the superiorization method (SM)—which perturbs a feasibility-seeking algorithm to reduce a secondary objective—fails to guarantee a superior objective value relative to the unperturbed feasibility-seeking run when using the Dynamic String-Averaging Projection (DSAP) framework with negative gradient perturbations. By analyzing the DSAP-based SM and establishing precise geometric and perturbation-resilience conditions, the authors show that, in certain settings, the SM can converge to a feasible point whose objective value does not meet the desired superiority, unless an inverse condition holds. The work highlights that guarantee results for SM must incorporate additional assumptions and provides practical guidance, such as initializing far enough from the feasible set to avoid the negative condition in large-scale applications. Overall, it clarifies the boundary of SM’s theoretical guarantees and offers actionable insight for practitioners aiming to leverage SM effectively in real-world problems.

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.

A necessary condition for the guarantee of the superiorization method

TL;DR

This paper identifies a negative condition under which the superiorization method (SM)—which perturbs a feasibility-seeking algorithm to reduce a secondary objective—fails to guarantee a superior objective value relative to the unperturbed feasibility-seeking run when using the Dynamic String-Averaging Projection (DSAP) framework with negative gradient perturbations. By analyzing the DSAP-based SM and establishing precise geometric and perturbation-resilience conditions, the authors show that, in certain settings, the SM can converge to a feasible point whose objective value does not meet the desired superiority, unless an inverse condition holds. The work highlights that guarantee results for SM must incorporate additional assumptions and provides practical guidance, such as initializing far enough from the feasible set to avoid the negative condition in large-scale applications. Overall, it clarifies the boundary of SM’s theoretical guarantees and offers actionable insight for practitioners aiming to leverage SM effectively in real-world problems.

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.

Paper Structure

This paper contains 7 sections, 5 theorems, 29 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Some applications that use the superiorization methodology
  3. Preliminaries: superiorization and dynamic string-averaging
  4. The superiorization methodology
  5. The convergence properties of the DSAP algorithm and its superiorized version
  6. The “ negative condition” on the superiorization method
  7. Conclusion

Key Result

Theorem 3.5

Assume that $\{C_{i}\}{}_{i=1}^{m}$ is a family of nonempty, closed and convex subsets of $X$ with a nonempty intersection $C$, satisfying Condition bound_reg. Let $\left\{ \beta_{k}\right\} _{k=0}^{\infty}$ be a sequence of non-negative numbers such that $\sum_{k=0}^{\infty}\beta_{k}<\infty$, let $ converges in the norm of $X$, and its limit belongs to $C$. That is, Algorithm alg:DSAP converges t

Theorems & Definitions (11)

  • Definition 3.1: cdh10
  • Theorem 3.5: cz12
  • Theorem 3.6: cz14-feje
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Corollary 4.3
  • proof
  • Remark 4.4
  • Example 4.5
  • ...and 1 more