The well-known Jahn-Graef-Younes algorithm, proposed by Jahn in 2006, generates all minimal elements of a finite set with respect to an ordering cone. It consists of two Graef-Younes procedures, namely the forward iteration, which eliminates a part of the non-minimal elements, followed by the backward iteration, which is applied to the reduced set generated by the previous iteration. Without using the backward iteration, we develop new algorithms that also compute all minimal elements of the initial set, by combining the forward iteration with certain sorting procedures based on cone-monotone functions. In particular, when the ordering cone is polyhedral, computational results obtained in MATLAB allow us to compare our algorithms with the Jahn-Graef-Younes algorithm, within a bi-objective optimization problem.
C. Günther and N. Popovici, New algorithms for discrete vector optimization based on the Graef-Younes method and cone-monotone sorting functions, Optimization, Volume 67, Issue 7, Pages 975-1003, 2018 (see https://www.tandfonline.com/doi/abs/10.1080/02331934.2018.1474469)