We study the Superiorization Methodology (SM) in the context of the General Dynamic
String-Averaging (GDSA) method in the inconsistent case (that is, where the
input operators don’t have a common fixed point) which primarily aims at achieving
convex feasibility while simultaneously reducing an objective function. In many scientific
and real-world problems modeled as constrained minimization tasks, striving for
the exact constrained optimum can be costly in terms of time, energy, and resources.
Therefore, applying the SM can offer a practical and efficient alternative. In particular,
we present a new “theorem of alternatives” for the superiorization method which leads
to investigation of theoretical conditions under which the superiorized version of the
GDSA algorithm converges to a “superior” feasible point, i.e., one with an objective
function value that is smaller or equal to that produced by the unperturbed feasibilityseeking
algorithm. While this question has only been partially addressed in the existing
literature, we present new sufficient conditions that guarantee that the SM attains such
a superior outcome.
Citation
Accepted for publication in Journal of Fixed Point Theory and Applications