A new theorem of alternatives leading to sufficient conditions for the superiorization guarantee question of Dynamic String-Averaging in the inconsistent case

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 do not 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 feasibility-seeking 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.