  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.
