We review the superiorization methodology, which can be thought of, in some cases, 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 an objective function value) to one returned by a feasibility-seeking only algorithm. We distinguish between two research directions in the superiorization methodology that nourish from the same general principle: Weak superiorization and strong superiorization and clarify their nature.
Preprint, September 30, 2014. Revised: November 27, 2014. Presented at the Tenth Workshop on Mathematical Modelling of Environmental and Life Sciences Problems, October 16-19, 2014, Constantza, Romania. http://www.ima.ro/workshop/tenth_workshop/.