Considerations in the Incremental Hypervolume Algorithm of the WFG

© 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.The hypervolume indicator (HV) has been subject of a lot of research in the last few years, mainly because its maximization yields near-optimal approximations of the Pareto optimal front of a multi-objective optimization problem. This feature has been exploited by several evolutionary optimizers, in spite of the considerable growth in computational cost that it is involved in the computation of HV as we increase the number of objectives. Some years ago, the Walking Fish Group (WFG) implemented a new version of the incremental hypervolume algorithm, named IWFG 1.01. This implementation is the fastest reported to date for determining the solution that contributes the least to the HV of a non-dominated set. Nevertheless, this new version has gone mostly unnoticed by the research community. We believe that this is due to an error in the source code provided by the authors of this algorithm, which appears when coupling it to a multi-objective evolutionary algorithm. In this paper, we describe this error, and we propose a solution to fix it. Moreover, we illustrate the significant gains in performance produced by IWFG 1.01 in many-objective optimization problems (i.e., problems having three or more objectives), when integrated into the S-Metric Selection Evolutionary Multi-Objective Algorithm (SMS-EMOA).

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