A Novel Method for Calculating the Parametric Hypervolume Indicator

Abstract

This paper presents a new methodology for calculating the hypervolume indicator (HVI)for multi-objective and parametric data. Existing multi-objective HVI calculation techniques cannot be directly used for parametric data because designers do not have preferences for parameters like they do for objectives. The novel method presented herein allows for the consideration of both objectives and parameters through the newly introduced hypercone heuristic (HCH). This heuristic relaxes the strict rules of parametric Pareto dominance for a more practical dominance assessment when comparing designs of differing parameter values without violating Pareto dominance rules. A parametric HVI (pHVI) enhances a design engineer’s toolkit by enabling both online and offline evaluation of parametric optimization results. The pHVI measure allows designers to compare solution sets, detect optimization convergence, and to better inform optimization procedures in a parametric context. Results show that the HCH-based pHVI yields a similar quality measure to the existing technique based on a support vector domain description (SVDD) in a fraction of the computational time. Furthermore, the novel HCH-based pHVI technique satisfies multi-objective HVI properties allowing previous applications of the HVI to be applied to multi-objective parametric optimization. This contribution enables the field of parametric optimization, and thus parametric design, to benefit from prior and future advances in the multi-objective optimization domain involving the HVI.