| Summary: | Deterioration of the searchability of Pareto dominance-based, many-objective evolutionary optimization algorithms is a well-known problem. Alternative solutions, such as scalarization-based and indicator-based approaches, have been proposed in the literature. However, Pareto dominance-based algorithms are still widely used. In this paper, we propose to redefine the calculation of Pareto-dominance. Instead of assigning solutions to non-dominated fronts, they are ranked according to the measure of dominating solutions referred to as <i>k</i>-Pareto optimality. In the case of probability measures, such re-definition results in an elegant and fast approximate procedure. Through experimental results on the many-objective <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>/</mo><mn>1</mn></mrow></semantics></math></inline-formula> knapsack problem, we demonstrate the advantages of the proposed approach: (1) the approximate calculation procedure is much faster than the standard sorting by Pareto dominance; (2) it allows for achieving higher hypervolume values for both multi-objective (two objectives) and many-objective (25 objectives) optimization; (3) in the case of many-objective optimization, the increased ability to differentiate between solutions results in a better compared to NSGA-II and NSGA-III. Apart from the numerical improvements, the probabilistic procedure can be considered as a linear extension of multidimentional topological sorting. It produces almost no ties and, as opposed to other popular linear extensions, has an intuitive interpretation.
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