Aggregated Partial Hypervolumes - An Overall Indicator for Performance Evaluation of Multimodal Multiobjective Optimization Methods

Multimodal multiobjective optimization (MMMOO) can be perceived as the combination of multiobjective optimization (MOO) and multimodal optimization (MMO). The performance of an MMMOO method should be thus assessed from both perspectives, leading to the prevalence of dual-metric indicators in the existing literature. This study first analyzes the ideal outcome of MMMOO for informed decision-making to determine the prerequisites of a theoretically and practically sound performance indicator. Then, it critically evaluates existing indicators, especially those that intend to measure success from the MMO perspective. Subsequently, it introduces Aggregated Partial Hypervolumes (APHVs) as a novel overall parametric performance indicator that not only addresses the drawbacks of existing ones but can also reflect the relative importance of MMO for the decision-maker. Finally, a few descriptive MMMOO examples are studied to verify that the optimal population according to APHVs matches our understanding of the ideal outcome of MMMOO, taking into account the relative importance of both the MMO and the MOO perspectives.