  The superiorization methodology is intended to work with input data of
constrained minimization problems, that is, a target function and a set of
constraints. However, it is based on an antipodal way of thinking to what leads
to constrained minimization methods. Instead of adapting unconstrained
minimization algorithms to handling constraints, it adapts feasibility-seeking
algorithms to reduce (not necessarily minimize) target function values. This is
done by inserting target-function-reducing perturbations into a
feasibility-seeking algorithm while retaining its feasibility-seeking ability
and without paying a high computational price. A superiorized algorithm that
employs component-wise target function reduction steps is presented. This
enables derivative-free superiorization (DFS), meaning that superiorization can
be applied to target functions that have no calculable partial derivatives or
subgradients. The numerical behavior of our derivative-free superiorization
algorithm is illustrated on a data set generated by simulating a problem of
image reconstruction from projections. We present a tool (we call it a
proximity-target curve) for deciding which of two iterative methods is "better"
for solving a particular problem. The plots of proximity-target curves of our
experiments demonstrate the advantage of the proposed derivative-free
superiorization algorithm.
