Life on the Edge: Characterising the Edges of Mutually Non-dominating Sets

Multi-objective optimisation yields an estimated Pareto front of mutually non- dominating solutions, but with more than three objectives, understanding the relationships between solutions is challenging. Natural solutions to use as landmarks are those lying near to the edges of the mutually non-dominating set. We propose four definitions of edge points for many-objective mutually non-dominating sets and examine the relations between them. The first defines edge points to be those that extend the range of the attainment surface. This is shown to be equivalent to finding points which are not dominated on projection onto subsets of the objectives. If the objectives are to be minimised, a further definition considers points which are not dominated under maximisation when projected onto objective subsets. A final definition looks for edges via alternative projections of the set. We examine the relations between these definitions and their efficacy in many dimensions for synthetic concave- and convex-shaped sets, and on solutions to a prototypical many-objective optimisation problem, showing how they can reveal information about the structure of the estimated Pareto front. We show that the "controlling dominance area of solutions" modification of the dominance relation can be effectively used to locate edges and interior points of high-dimensional mutually non-dominating sets.