Cell Mapping Methods

Sun, Jian‐Qiao; Xiong, Fu-Rui; Schütze, Oliver; Hernández, C. Rodríguez

doi:10.1007/978-981-13-0457-6

Cited by 36 publications

(16 citation statements)

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“…Hence, it is natural that the candidate solution set (i.e., the set computed by a given solver) is not restricted to finitely many points, but may also form a continuous set. This is in fact already the case for set-based optimization techniques such as the cell-to-cell mappings ( [15][16][17]) and the subdivision techniques ( [10,18,19]). In the context of evolutionary multi-objective optimization, typically a finite set of candidate solutions (a population) is generated ( [20][21][22][23]).…”

Section: Introductionmentioning

confidence: 92%

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: Introductionmentioning

confidence: 92%

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

Bogoya

Vargas

Cuate

et al. 2018

MCA

Self Cite

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“…All of these methods typically yield high convergence rates but are, in turn, of local nature. A possible alternative is given by set oriented methods such as subdivision and cell mapping techniques [53,54] and evolutionary algorithms [55][56][57][58][59] that are of global nature and are capable of computing a finite size approximation of the Pareto front in one single run.…”

Section: Multi-objective Optimizationmentioning

confidence: 99%

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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“…The Simple Cell Mapping (SCM) method is usually used to estimate the DOAs of different passive biped walkers [6,7,10]. The SCM method is an effective way to study the global stability of nonlinear dynamical systems, and its basic idea is to obtain the global properties of system by using the ordered information of local finite cells rather than investigating the properties of all states in the infinite state space [11][12][13]. However, there are still some issues to be solved when estimating the DOA of passive biped walker by the SCM method.…”

Section: Introductionmentioning

confidence: 99%

“…For making some concepts clearly, the basic procedures of the SCM method are introduced as follows [11,13]. Firstly, a feasible continuous domain with boundary is chosen from the infinite state space as the Domain Of Interest (DOI).…”

Section: Introductionmentioning

confidence: 99%

“…Dellnitz et al introduced the subdivision algorithm for obtaining the invariant sets of nonlinear dynamical systems [17]. This algorithm can be integrated with most cell mapping methods to solve different problems, such as the multiobjective optimization problems [18] and the high-dimensional problems [13]. Although both the subdivision algorithm and the proposed method in this paper use the iterative method to decrease the number of cell mappings, they have different goals.…”

Section: Introductionmentioning

confidence: 99%

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An Improved Method for Estimating the Domain of Attraction of Passive Biped Walker

Wang

Cao

Jinlin

2019

Discrete Dynamics in Nature and Society

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An indicator of a passive biped walker’s global stability is its domain of attraction, which is usually estimated by the simple cell mapping method. It needs to calculate a large number of cells’ Poincare mapping result in the estimating process. However, the Poincare mapping is usually computationally expensive and time-consuming due to the complex dynamical equation of the passive biped walker. How to estimate the domain of attraction efficiently and reliably is a problem to be solved. Based on the simple cell mapping method, an improved method is proposed to solve it. The proposed method uses the multiple iteration algorithm to calculate a stable domain of attraction and effectively decreases the total number of Poincare mappings. Through the simulation of the simplest passive biped walker, the improved method can obtain the same domain of attraction as that calculated using the simple cell mapping method and reduce calculation time significantly. Furthermore, this improved method not only proposes a way of rapid estimating the domain of attraction, but also provides a feasible tool for selecting the domain of interest and its discretization level.

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Cell Mapping Methods

Cited by 36 publications

References 0 publications

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

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

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

An Improved Method for Estimating the Domain of Attraction of Passive Biped Walker

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