A Weakly Pareto Compliant Quality Indicator

Dilettoso, E.; Rizzo, Santi Agatino

doi:10.3390/mca22010025

Cited by 12 publications

(7 citation statements)

References 37 publications

(52 reference statements)

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“…In this section, we study the behavior of I GD p,q , I IGD p,q , and I ∆ p,q as performance indicators. An example of a weakly Pareto-compliant performance indicator is the Degree of Approximation (DOA; see Reference [10]).…”

Section: Thementioning

confidence: 99%

“…The Hausdorff distance d H (e.g., Reference [1]) measures how far two subsets of a metric space are from each other. Due to its properties, it is frequently used in many research areas such as computer vision [2][3][4], fractal geometry [5], the numerical computation of attractors in dynamical systems [6][7][8], or convergence of multi-objective algorithms to the Pareto set/front of a given multi-objective optimization problem [9][10][11][12][13][14][15]. One possible drawback of the classical Hausdorff distance, however, is that it punishes single outliers which leads to inequitable performance evaluations in some cases.…”

Section: Introductionmentioning

confidence: 99%

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The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review

2019

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A brief but comprehensive review of the averaged Hausdorff distances that have recentlybeen introduced as quality indicators in multi-objective optimization problems (MOPs) is presented.First, we introduce all the necessary preliminaries, definitions, and known properties of thesedistances in order to provide a stat-of-the-art overview of their behavior from a theoretical pointof view. The presentation treats separately the definitions of the (p, q)-distances GDp,q, IGDp,q, and Δp,q for finite sets and their generalization for arbitrary measurable sets that covers as an importantexample the case of continuous sets. Among the presented results, we highlight the rigorousconsideration of metric properties of these definitions, including a proof of the triangle inequalityfor distances between disjoint subsets when p, q ≥ 1, and the study of the behavior of associatedindicators with respect to the notion of compliance to Pareto optimality. Illustration of these resultsin particular situations are also provided. Finally, we discuss a collection of examples and numericalresults obtained for the discrete and continuous incarnations of these distances that allow for anevaluation of their usefulness in concrete situations and for some interesting conclusions at the end,justifying their use and further study.

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Section: Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review

2019

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“…Unlike evolutionary algorithms for single objective optimization problems (SOPs), maintaining diversity in decision space is not a priority for most MOEAs; most of the performance indicators are developed in order to measure the accuracy based only on the objective function (e.g., the hypervolume [13] and the Degree of Approximation [14]). As exceptions, we have some of the measures used for multimodal optimization (see [15]) and two particular examples.…”

Section: Background and Related Workmentioning

confidence: 99%

Variation Rate to Maintain Diversity in Decision Space within Multi-Objective Evolutionary Algorithms

Cuate

Schütze²

2019

MCA

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The performance of a multi-objective evolutionary algorithm (MOEA) is in most cases measured in terms of the populations’ approximation quality in objective space. As a consequence, most MOEAs focus on such approximations while neglecting the distribution of the individuals of their populations in decision space. This, however, represents a potential shortcoming in certain applications as in many cases one can obtain the same or very similar qualities (measured in objective space) in several ways (measured in decision space). Hence, a high diversity in decision space may represent valuable information for the decision maker for the realization of a given project. In this paper, we propose the Variation Rate, a heuristic selection strategy that aims to maintain diversity both in decision and objective space. The core of this strategy is the proper combination of the averaged distance applied in variable space together with the diversity mechanism in objective space that is used within a chosen MOEA. To show the applicability of the method, we propose the resulting selection strategies for some of the most representative state-of-the-art MOEAs and show numerical results on several benchmark problems. The results demonstrate that the consideration of the Variation Rate can greatly enhance the diversity in decision space for all considered algorithms and problems without a significant loss in the approximation qualities in objective space.

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“…The design of a standard mechanism to generate welldiversified reference point sets for benchmark problems in EMO, varying the geometry, dimension, and cardinality. These reference point sets are critical for the calculation of some QIs [7,[16][17][18] and to construct selection mechanisms of multi-objective evolutionary algorithms (MOEAs) [19][20][21]. 2.…”

Section: Introductionmentioning

confidence: 99%