Multiplicative approximations and the hypervolume indicator

Indicator-based algorithms have become a very popular approach to solve multi-objective optimization problems. In this paper, we contribute to the theoretical understanding of algorithms maximizing the hypervolume for a given problem by distributing µ points on the Pareto front. We examine this common approach with respect to the achieved multiplicative approximation ratio for a given multi-objective problem and relate it to a set of µ points on the Pareto front that achieves the best possible approximation ratio. For the class of linear fronts and a class of concave fronts, we prove that the hypervolume gives the best possible approximation ratio. In addition, we examine Pareto fronts of different shapes by numerical calculations and show that the approximation computed by the hypervolume may differ from the optimal approximation ratio. Categories and Subject Descriptors: F.2 [Theory of Computation]: Analysis of Algorithms and Problem Complexity General Terms: Theory, Algorithms,...

• Type: Conference
• Year: 2009
• Where: GECCO
• Authors: Tobias Friedrich, Christian Horoba, Frank Neumann
• Keywords: Approximation Ratio, Artificial Intelligence, GECCO 2009, Pareto Fronts, Possible Approximation Ratio