  The superiorization methodology can be thought of as lying conceptually
between feasibility-seeking and constrained minimization. It is not trying to
solve the full-fledged constrained minimization problem composed from the
modeling constraints and the chosen objective function. Rather, the task is to
find a feasible point which is "superior" (in a well-defined manner) with
respect to the objective function, to one returned by a feasibility-seeking
only algorithm. We telegraphically review the superiorization methodology and
where it stands today and propose a rigorous formulation of its, yet only
partially resolved, guarantee problem. The real-world situation in an
application field is commonly represented by constraints defined by the
modeling process and the data, obtained from measurements or otherwise dictated
by the model-user. The feasibility-seeking problem requires to find a point in
the intersection of all constraints without using any objective function to aim
at any specific feasible point. At the heart of the superiorization methodology
lies the modeler desire to use an objective function, that is exogenous to the
constraints, in order to seek a feasible solution that will have lower (not
necessarily minimal) objective function value. This aim is less demanding than
full-fledged constrained minimization but more demanding than plain
feasibility-seeking. Putting emphasis on the need to satisfy the constraints,
because they represent the real-world situation, one recognizes the "asymmetric
roles of feasibility-seeking and objective function reduction", namely, that
fulfilling the constraints is the main task while reduction of the exogenous
objective function plays only a secondary role. There are two research
directions in the superiorization methodology: Weak superiorization and strong
superiorization.
