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

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.Comment: 15 pages, 21 figures, 3 table

Advanced OR and AI Methods in Transportation PATH RELINKING FOR MULTIPLE OBJECTIVE COMBINATORIAL OPTIMIZATION. TSP CASE STUDY

Abstract. The paper presents a new metaheuristic algorithm for multiple objective combinatorial optimization based on the idea of path relinking. The algorithm is applied to the traveling salesperson problem with multiple link (arc) costs, corresponding to multiple objectives. The multiple costs may for example correspond to financial cost of travel along a link, time of travel, or risk in case of hazardous materials. The algorithm searches for new good solutions along paths in the decision space connecting two other good solutions. It is compared experimentally to state of the art algorithms for multiple objective TSP. 1

problem and the Pareto

A comparative study of multiple-objectiv

Evaluating the quality of approximations to the non-dominated set

: The growing interest in hard multiple objective combinatorial and non-linear problems resulted in a significant number of heuristic methods aiming at generating sets of feasible solutions as approximations to the set of non-dominated solutions. The issue of evaluating these approximations is addressed. Such evaluations are useful when performing experimental comparisons of different multiple objective heuristic algorithms, when defining stopping rules of multiple objective heuristic algorithms, and when adjusting parameters of heuristic algorithms to a given problem. A family of outperformance relations that can be used to compare approximations under very weak assumptions about a decision-maker's preferences is introduced. These outperformance relations define incomplete orders in the set of all approximations. It is shown that in order to compare approximations, which are incomparable according to the outperformance relations, much stronger assumptions about the decision-maker's p..