Strict Fej\'er Monotonicity by Superiorization of Feasibility-Seeking Projection Methods

TL;DR

This paper demonstrates that superiorized projection methods for convex feasibility problems either converge to an optimal solution or exhibit strict Fejér monotonicity, providing new insights into their mathematical behavior.

Contribution

It establishes that superiorized dynamic string-averaging projection algorithms either find a solution to the constrained minimization or are strictly Fejér monotone, enhancing understanding of their convergence properties.

Findings

Sequences converge to feasible points.

Sequences are either optimal or strictly Fejér monotone.

Provides new mathematical insights into superiorization methodology.

Abstract

We consider the superiorization methodology, which can be thought of as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full fledged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to the objective function value) to one returned by a feasibility-seeking only algorithm. Our main result reveals new information about the mathematical behavior of the superiorization methodology. We deal with a constrained minimization problem with a feasible region, which is the intersection of finitely many closed convex constraint sets, and use the dynamic string-averaging projection method, with variable strings and variable weights, as a feasibility-seeking algorithm. We show that any sequence, generated by the superiorized version of a dynamic string-averaging projection algorithm, not only converges to a feasible point but, additionally, either its limit point solves the constrained minimization problem or the sequence is strictly Fej\'er monotone with respect to a subset of the solution set of the original problem.