The Hypervolume Indicator: Computational Problems and Algorithms

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 article 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 article 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.