ND-Tree-Based Update: A Fast Algorithm for the Dynamic Nondominance Problem

In this paper, we propose a new method called ND-Tree-based update (ND-Tree) for the dynamic nondominance problem, i.e., the problem of online update of a Pareto archive composed of mutually nondominated points. It uses a new ND-Tree data structure in which each node represents a subset of points contained in a hyperrectangle defined by its local approximate ideal and nadir points. By building subsets containing points located close in the objective space and using basic properties of the local ideal and nadir points we can efficiently avoid searching many branches in the tree. ND-Tree may be used in multiobjective evolutionary algorithms and other multiobjective metaheuristics to update an archive of potentially nondominated points. We prove that the proposed algorithm has sublinear time complexity under mild assumptions. We experimentally compare ND-Tree to the simple list, Quad-tree, and M-Front methods using artificial and realistic benchmarks with up to ten objectives and show that with this new method substantial reduction of the number of point comparisons and computational time can be obtained. Furthermore, we apply the method to the nondominated sorting problem showing that it is highly competitive to some recently proposed algorithms dedicated to this problem.