  In this paper we study the split minimization problem that consists of two
constrained minimization problems in two separate spaces that are connected via
a linear operator that maps one space into the other. To handle the data of
such a problem we develop a superiorization approach that can reach a feasible
point with reduced (not necessarily minimal) objective function values. The
superiorization methodology is based on interlacing the iterative steps of two
separate and independent iterative processes by perturbing the iterates of one
process according to the steps dictated by the other process. We include in our
developed method two novel elements. The first one is the permission to restart
the perturbations in the superiorized algorithm which results in a significant
acceleration and increases the computational efficiency. The second element is
the ability to independently superiorize subvectors. This caters to the needs
of real-world applications, as demonstrated here for a problem in
intensity-modulated radiation therapy treatment planning.
