The Hypervolume Indicator: Problems and Algorithms

TL;DR

This paper reviews the computational problems related to the hypervolume indicator in multiobjective optimization, discussing algorithms, complexities, and providing guidance for selecting suitable implementations.

Contribution

It offers a comprehensive overview of hypervolume-related computational problems, their relationships, and the algorithms used, aiding future research and practical application.

Findings

Summarizes the main algorithms for hypervolume computation.

Analyzes the asymptotic complexity and runtime of algorithms.

Provides guidance on selecting the most appropriate algorithms.

Abstract

The hypervolume indicator is one of the most used set-quality indicators for the assessment of stochastic multiobjective optimizers, as well as for selection in evolutionary multiobjective optimization algorithms. Its theoretical properties justify its wide acceptance, particularly the strict monotonicity with respect to set dominance which is still unique of hypervolume-based indicators. This paper discusses the computation of hypervolume-related problems, highlighting the relations between them, providing an overview of the paradigms and techniques used, a description of the main algorithms for each problem, and a rundown of the fastest algorithms regarding asymptotic complexity and runtime. By providing a complete overview of the computational problems associated to the hypervolume indicator, this paper serves as the starting point for the development of new algorithms, and supports users in the identification of the most appropriate implementations available for each problem.