Multi-objective Neutral Neighbors': What could be the definition(s)?

There is a significant body of research on neutrality and its effects in single-objective optimization. Particularly, the neutrality concept has been precisely defined and the neutrality between neighboring solutions efficiently exploited in local search algorithms. The extension of neutrality to multi-objective optimization is not straightforward and its effects on the dynamics of multi-objective optimization methods are not clearly understood. In order to develop strategies to exploit neutral neighbors in multi-objective local search algorithms, it is important and necessary to clearly define neutrality in the multi-objective context. In this paper, we propose several definitions of the neutrality property between neighboring solutions. A natural definition comes from the Pareto-dominance, widely used in multi-objective optimization. In addition, definitions derived from epsilon and hypervolume indicators are also proposed as such indicators are usually used to compare sets of solutions. We analyze permutation problems under the proposed definitions of neutrality and show that each definition of neutrality leads to a particular structure of the problem.