Abstract
With many efficient solutions for a multi-objective optimization problem, this paper aims to cluster the Pareto Front in a given number of clusters K and to detect isolated points. K-center problems and variants are investigated with a unified formulation considering the discrete and continuous versions, partial K-center problems, and their min-sum-K-radii variants. In dimension three (or upper), this induces NP-hard complexities. In the planar case, common optimality property is proven: non-nested optimal solutions exist. This induces a common dynamic programming algorithm running in polynomial time. Specific improvements hold for some variants, such as K-center problems and min-sum K-radii on a line. When applied to N points and allowing to uncover M<N points, K-center and min-sum-K-radii variants are, respectively, solvable in O(K(M+1)NlogN) and O(K(M+1)N2) time. Such complexity of results allows an efficient straightforward implementation. Parallel implementations can also be designed for a practical speed-up. Their application inside multi-objective heuristics is discussed to archive partial Pareto fronts, with a special interest in partial clustering variants.
Highlights
This paper is motivated by real-world applications of multi-objective optimization (MOO)
The complexity for 2D Pareto front (PF) cases is very similar to the 1D cases; the 2D PF extension does not induce major difficulties in terms of complexity results. 2D PF cases may induce significant differences compared to the general 2D cases
The p-center problems are NP-hard in a planar Euclidean space [17], since adding the PF hypothesis leads to the polynomial complexity of Theorem 1, which allows for an efficient, straightforward implementation of the algorithm
Summary
This paper is motivated by real-world applications of multi-objective optimization (MOO). Several non-dominated points in the objective space can be generated, defining efficient solutions, which are the best compromises. MOO approaches may generate large sets of efficient solutions using Pareto dominance [3]. Summarizing the shape of a PF may be required for presentation to decision makers In such a context, clustering problems are useful to support decision making to present a view of a PF in clusters, the density of points in the cluster, or to select the most central cluster points as representative points. Note than similar problems are of interest for population MOO heuristics such as evolutionary algorithms to archive representative points of a partial Pareto fronts, or in selecting diversified efficient solutions to process mutation or cross-over operators [4,5]
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