Why Dominance is Not Enough: Lessons from Practical Evolutionary Multi-Objective Algorithms

Practical EMO algorithms like NSGA-II, NSGA-III, and SMS-EMOA combine the dominance relation with diversity criteria to identify promising solutions. Despite many success stories, their theoretical foundation remains underdeveloped, with key questions still unanswered—such as which information obtained throughout the evolution is critical for their success. We explore the limitations of the information provided by the dominance relation between search points encountered so far. We construct an artificial problem with a small Pareto set where almost all pairs of search points are incomparable. For this problem, we prove that any black-box EMO algorithm that only relies on the dominance relation for making decisions and only use variation operators that are invariant to bit values, fails spectacularly, requiring exponential time with high probability. In stark contrast, NSGA-II, NSGA-III, and SMS-EMOA efficiently cover the Pareto front in expected quadratic time by incorporating additional information, such as objective values. Our results highlight the superiority of practical EMO algorithms and the necessity of using information beyond dominance for effective multi-objective optimisation.