On cone based decompositions of proper Pareto optimality

In recent years, the research focus in multi-objective optimization has shifted from approximating the Pareto optimal front in its entirety to identifying solutions that are well-balanced among their objectives. Proper Pareto optimality is an established concept for eliminating Pareto optimal solutions that exhibit unbounded tradeoffs. Imposing a strict tradeoff bound allows specifying how many units of one objective one is willing to trade in for obtaining one unit of another objective. This notion can be translated to a dominance relation, which we denote by M-domination. The mathematical properties of M-domination are thoroughly analyzed in this paper yielding key insights into its applicability as decision making aid. We complement our work by providing four different geometrical descriptions of the M-dominated space given by a union of polyhedral cones. A geometrical description does not only yield a greater understanding of the underlying tradeoff concept, but also allows a quantification of the space dominated by a particular solution or an entire set of solutions. These insights enable us to formulate volume-based approaches for finding approximations of the Pareto front that emphasize regions that are well-balanced among their tradeoffs in subsequent works.