Efficient Dominance Filtering for Unions and Minkowski Sums of Non-Dominated Sets

Abstract

The repeated filtering of vectors for non-dominance is an important component in many multi-objective programming approaches, like e.g. decomposition approaches, dynamic programming or meta heuristics. Often the set of vectors to be filtered is given as the union A ∪ B or Minkowski sum A + B of Pareto (or stable) sets, i.e. within both sets A and B the vectors are pairwise non-dominated. We propose several algorithms for both problems and compare them to a well-known static divide-and-conquer non-dominance filtering algorithm. Based on numerical experiments, we give recommendations for choosing a suitable method for particular situations, depending on, e.g., the number of objectives or the relative sizes of the sets A and B. Moreover, we consider non-dominance filtering for multi-set sums S = A1 + ... + A s .

Keywords: dominance filtering, ND-trees, dynamic programming, multi-objective optimization, decomposition