Computing minimal elements of finite families of sets w.r.t. preorder relations in set optimization

We propose new algorithms for computing all minimal elements of a nonempty finite family of sets in a real linear space, with respect to a preorder relation defined on the power set of that space. These algorithms are based on a set-valued counterpart of the well-known Graef-Younes reduction procedure, originally conceived for vector optimization. One of our algorithms consists of two subsequent (forward-backward) reduction procedures, similarly to the classical Jahn-Graef-Younes method. Another algorithm involves a pre-sorting procedure with respect to a strongly increasing real-valued function, followed by a single (forward) reduction procedure. Numerical experiments in MATLAB allow us to compare our algorithms for special test families of line segments with respect to `-type, u-type and s-type preorder relations, currently used in set optimization.