  Superiorization reduces, not necessarily minimizes, the value of a target
function while seeking constraints-compatibility. This is done by taking a
solely feasibility-seeking algorithm, analyzing its perturbations resilience,
and proactively perturbing its iterates accordingly to steer them toward a
feasible point with reduced value of the target function. When the perturbation
steps are computationally efficient, this enables generation of a superior
result with essentially the same computational cost as that of the original
feasibility-seeking algorithm. In this work, we refine previous formulations of
the superiorization method to create a more general framework, enabling target
function reduction steps that do not require partial derivatives of the target
function. In perturbations that use partial derivatives the step-sizes in the
perturbation phase of the superiorization method are chosen independently from
the choice of the nonascent directions. This is no longer true when
component-wise perturbations are employed. In that case, the step-sizes must be
linked to the choice of the nonascent direction in every step. Besides
presenting and validating these notions, we give a computational demonstration
of superiorization with component-wise perturbations for a problem of
computerized tomography image reconstruction.
