The Hypervolume Indicator Revisited: On the Design of Pareto-compliant Indicators Via Weighted Integration

By Eckart Zitzler, Dimo Brockhoff, Lothar Thiele. International Conference on Evolutionary Multi-Criterion Optimization, 2007

669 citations recorded by the source corpus.

The design of quality measures for approximations of the Pareto-optimal set is of high importance not only for the performance assessment, but also for the construction of multiobjective optimizers. Various measures have been proposed in the literature with the intention to capture different preferences of the decision maker. A quality measure that possesses a highly desirable feature is the hypervolume measure: whenever one approximation completely dominates another approximation, the hypervolume of the former will be greater than the hypervolume of the latter. Unfortunately, this measure--as any measure inducing a total order on the search space--is biased, in particular towards convex, inner portions of the objective space. Thus, an open question in this context is whether it can be modified such that other preferences such as a bias towards extreme solutions can be obtained. This paper proposes a methodology for quality measure design based on the hypervolume measure and demonstrates its usefulness for three types of preferences.

Subfields: Computational Theory and Mathematics, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ae4c3d6d5959b13a78a9027b0bf7a03495e6a05b

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Citations in the corpus

Pymoo: Multi-Objective Optimization in Python (2020, background): “It has been shown that Hypervolume is Pareto compliant [55].”
Performance indicators in multiobjective optimization (2018, background): “More recently, a pondered hyper-volume by weights was introduced by [62] to give a preference of an objective according to another.”
HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization (2011, positive): “Besides the names, there are also different deﬁnitions available, based on polytopes (Zitzler and Thiele, 1999), the Lebesgue measure (Lau-manns et al., 1999; Knowles, 2002; Fleischer, 2003), or the attainment function (Zitzler et al., 2007).”
Modified Distance Calculation in Generational Distance and Inverted Generational Distance (2015, background): “It has been repeatedly pointed out in the literature [14], [15], [18], [24], [26] that Pareto non-compliant misleading results can be obtained from some other performance indicators.”
Performance metrics in multi-objective optimization (2015, background): “It has been widely accepted since it offers the following unique and desirable properties [56]: i) whenever one approximation set completely dominates another approximation set, the hypervolume of the former will be greater than the hypervolume of the latter.”
Evolutionary many-objective optimization: A short review (2008, background): “Several variants of IBEAs have been proposed in the literature [44]-[46].”
A Multiobjective Evolutionary Algorithm Using Gaussian Process-Based Inverse Modeling (2015, background): “solution sets, has most widely been adopted [21]–[24].”
Automatically improving the anytime behaviour of optimisation algorithms (2014, background): “[61] suggest to model this preference by considering the following weight function (adapted here to minimisation):”
Quality Assessment of Pareto Set Approximations (2008, background): “Note: preference information can be incorporated into the hypervolume indicator, so that more emphasis can be placed on certain parts of the Pareto front than others (e.g. the middle, the extremes, etc.), whilst maintaining monotonicity (Zitzler et al., 2007).”
An easy-to-use real-world multi-objective optimization problem suite (2020, background): “HV is the only Pareto-compliant quality indicator known so far in the literature [51].”
Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point (2009, background): “These developments motivate why several researchers have been trying to better understand the hypervolume indicator from a theoretical perspective, e.g., [11, 28, 23].”