A Generalization of the Averaged Hausdorff Distance

Vargas, Andrés; Bogoya, J. M.

doi:10.13053/cys-22-2-2950

Cited by 7 publications

(13 citation statements)

References 10 publications

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“…In particular, this is a natural setting in multi-objective optimization because the solution of such a problem typically forms a set of certain dimension (and is thus not given by finitely many points). We have shown that the extended indicators keep the nice metric properties from its finite-version predecessors (see [14,26]). Moreover, for GD p,q , sufficient conditions have been provided ensuring that certain compliance to Pareto optimality of this indicator can be guaranteed.…”

Section: Discussionmentioning

confidence: 92%

“…With the aid of Theorem 1 we generalize the results of [26] (Section 3). For easy reference, we provide here slightly abbreviated but complete proofs.…”

Section: Definition Of ∆ Pq For Measurable Setsmentioning

confidence: 85%

“…The extension of ∆ p,q to measurable sets given in Definition 6 preserves the nice metric properties of the finite version considered in [26] (Section 3). In particular, Theorem 1 enables us to show a result analogous to [26] (Theorem 3.3).…”

Section: Metric Propertiesmentioning

confidence: 98%

“…Definition 4 (Vargas-Bogoya [26]). For p, q ∈ R × , and finite sets A, B ⊂ R n the value For finite sets, the indicator ∆ p,q , introduced in [26] was also defined for p or q = 0, and even for p or q = ±∞. It is a generalization of ∆ p in the sense that between disjoint subsets we have…”

Section: (V)mentioning

confidence: 99%

“…In [26], a modification of the ∆ p indicator called the (p, q)-averaged Hausdorff distance ∆ p,q has been introduced by the first two authors. This indicator generalizes the averaged Hausdorff distance ∆ p , is strongly related to the Hausdorff distance d H , and admits an expression in terms of the matrix p,q -norm · p,q .…”

Section: Introductionmentioning

confidence: 99%

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A (p,q)-Averaged Hausdorff Distance for Arbitrary Measurable Sets

Bogoya

Vargas

Cuate

et al. 2018

MCA

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The Hausdorff distance is a widely used tool to measure the distance between different sets. For the approximation of certain objects via stochastic search algorithms this distance is, however, of limited use as it punishes single outliers. As a remedy in the context of evolutionary multi-objective optimization (EMO), the averaged Hausdorff distance Δ p has been proposed that is better suited as an indicator for the performance assessment of EMO algorithms since such methods tend to generate outliers. Later on, the two-parameter indicator Δ p , q has been proposed for finite sets as an extension to Δ p which also averages distances, but which yields some desired metric properties. In this paper, we extend Δ p , q to a continuous function between general bounded subsets of finite measure inside a metric measure space. In particular, this extension applies to bounded subsets of R k endowed with the Euclidean metric, which is the natural context for EMO applications. We show that our extension preserves the nice metric properties of the finite case, and finally provide some useful numerical examples that arise in EMO.

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Section: Discussionmentioning

confidence: 92%

“…With the aid of Theorem 1 we generalize the results of [26] (Section 3). For easy reference, we provide here slightly abbreviated but complete proofs.…”

Section: Definition Of ∆ Pq For Measurable Setsmentioning

confidence: 85%

Section: Metric Propertiesmentioning

confidence: 98%

Section: (V)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A (p,q)-Averaged Hausdorff Distance for Arbitrary Measurable Sets

Bogoya

Vargas

Cuate

et al. 2018

MCA

Self Cite

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Explicit multiobjective model predictive control for nonlinear systems under uncertainty

Castellanos

Ober‐Blöbaum

Peitz

2020

Intl J Robust & Nonlinear

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In real-world problems, uncertainties (eg, errors in the measurement, precision errors, among others) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems, where in the case of uncertainties, optimal solutions can degrade in quality or they can even become unfeasible. Thus, there is the need to design methods that can handle uncertainty. In this work, we consider nonlinear multiobjective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control. For such problems, not much has been reported in terms of uncertainties. To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. In order to reduce the numerical cost of the offline phase-which grows exponentially with the parameter dimension-we exploit symmetries in the control problems. Furthermore, in order to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multiobjective optimization problem. We test our framework on a car maneuvering problem where safety and speed are the objectives. The multiobjective framework allows for online adaptations of the desired objective. Our results show that the method is capable of designing driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.

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