ND-Tree-based update: a Fast Algorithm for the Dynamic Non-Dominance Problem

TL;DR

The paper introduces ND-Tree, a novel data structure and algorithm for efficiently updating Pareto archives in multiobjective optimization, significantly reducing computational time compared to existing methods.

Contribution

It presents the ND-Tree data structure and algorithm, enabling fast, sub-linear time updates for dynamic non-dominance problems in multiobjective optimization.

Findings

ND-Tree reduces point comparison counts.

It achieves faster update times than existing methods.

The approach is effective for high-dimensional objectives.

Abstract

In this paper we propose a new method called ND-Tree-based update (or shortly ND-Tree) for the dynamic non-dominance problem, i.e. the problem of online update of a Pareto archive composed of mutually non-dominated 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 non-dominated points. We prove that the proposed algorithm has sub-linear 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 10 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 non-dominated sorting problem showing that it is highly competitive to some recently proposed algorithms dedicated to this problem.