On cone-based decompositions of proper Pareto-optimality in multi-objective optimization

In recent years, 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 in a classical definition of proper Pareto optimality allows specifying how many units of one objective one is willing to trade in for obtaining one unit of another objective. Recent studies have shown that this notion shows favorable convergence properties. One of the aims of this paper is to translate the proper Pareto optimality notion to an ordering 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 and in designing population-based algorithms for solving multi-objective optimization problems. 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 hypervolume dominated by a particular solution or an entire set of solutions. These descriptions shall enable researchers to formulate hypervolume-based approaches for finding approximations of the Pareto front that emphasize regions that are well-balanced among their tradeoffs in subsequent works.