Obtaining the efficient set of nonlinear biobjective optimization problems via interval branch-and-bound methods

Fernández, José María Pérez; G.-Tóth, Boglárka

doi:10.1007/s10589-007-9135-8

Cited by 41 publications

(41 citation statements)

References 36 publications

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“…The methods proposed in the literature with that purpose are specialized either in particular problems or for a particular type of multiobjective problems. To the extent of our knowledge, only two exact general methods, namely, two interval branch-and-bound methods (see [11,19]) have been proposed in literature which obtain an enclosure of those sets up to a prespecified precision. Specifically, they offer a list of boxes (multidimensional intervals) whose union contains the complete efficient set (and their images the corresponding Pareto-front) as a solution.…”

Section: Definitionmentioning

confidence: 99%

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A Triobjective Model for Locating a Public Semiobnoxious Facility in the Plane

Fernández

Redondo

Arrondo

et al. 2015

Mathematical Problems in Engineering

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A new mathematical model for locating a single semiobnoxious facility in the plane is proposed. Three objectives are taken into consideration. The first one maximizes the efficiency of the service provided by the facility to some users, by minimizing the sum of weighted distances between the facility and those users. The second one minimizes the social cost caused by the undesirable effects produced by the facility, by minimizing the sum of the repulsions of the affected people (as they feel it). The third one aims to distribute the repulsions fairly (as equal as possible) among the affected people. To prove that the new model can be tackled in practice, two recent general-purpose multiobjective evolutionary algorithms, MOEA/D and FEMOEA, are suggested to obtain a discrete approximation of its Pareto-front. A computational study shows that both algorithms are suitable to cope with the problem.

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Section: Definitionmentioning

confidence: 99%

“…In this work we restrict ourselves to continuous problems, in which the facility can be located in any place within a given subset of the plane. Although most multiobjective location papers deal with discrete or network problems, we can find many papers on continuous problems as well (see, for instance [10][11][12], to name a few).…”

Section: Introductionmentioning

confidence: 99%

A Triobjective Model for Locating a Public Semiobnoxious Facility in the Plane

Fernández

Redondo

Arrondo

et al. 2015

Mathematical Problems in Engineering

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“…It turns out that if we appropriately take this uncertainty into account, then (verified) algorithms for computing the resulting Pareto set become possible. Such algorithms were described, for the case of n = 2 objective functions f j defined on bounded subsets of R m , in Tóth and Fernández, 2006;2007;2009). In this paper, we extend these algorithms to the general case of arbitrary computable objective functions defined on a general computable set X.…”

Section: Introductionmentioning

confidence: 99%

Verified Methods for Computing Pareto Sets: General Algorithmic Analysis

G.-Tóth¹,

Kreinovich²

2009

International Journal of Applied Mathematics and Computer Science

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In many engineering problems, we face multi-objective optimization, with several objective functions f1, . . . , fn. We want to provide the user with the Pareto set-a set of all possible solutions x which cannot be improved in all categories (i.e., for which fj (x ) ≥ fj (x) for all j and fj(x ) > fj(x) for some j is impossible). The user should be able to select an appropriate trade-off between, say, cost and durability. We extend the general results about (verified) algorithmic computability of maxima locations to show that Pareto sets can also be computed.

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“…However, for a majority of MOPs, it is not easy to obtain an exact description of the efficient set or Pareto-front, since those sets typically include an infinite number of points (usually a continuum set). To the extent of our knowledge, only two exact general methods, namely, two interval branch-and-bound methods (see [25,26]) have been proposed in literature which obtain an enclosure of those sets up to a pre-specified precision. Specifically, they offer a list of boxes (multidimensional intervals) whose union contains the complete efficient set (and their images the corresponding Pareto-front) as a solution.…”

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confidence: 99%

“…On the other hand, there are many real world problems where only two (maybe three) objectives are considered. As an example, the authors are familiar with continuous location theory, an area of operations research where only bi-objective problems have been considered so far (see [25,26] and the references therein). Location theory is a prolific research area, to the point that it has now its own entry (90B85) in the Mathematics Subject Classification used by Mathematical Reviews and Zentralblatt für Mathematic.…”

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confidence: 99%

FEMOEA: a fast and efficient multi-objective evolutionary algorithm

2016

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A multi-objective evolutionary algorithm which can be applied to many nonlinear multi-objective optimization problems is proposed. Its aim is to quickly obtain a fixed size set approximating the complete Pareto-front. It adapts ideas from different multi-objective optimization evolutionary algorithms, but also incorporates new devices. In particular, the search in the space is carried out on promising areas (hyperspheres) determined by a radius value, which decreases as the optimization procedure evolves. This mechanism helps to maintain a balance between exploration and exploitation of the search space. Additionally, a new local search method which accelerates the convergence of the population towards the optimal Pareto-front, has been incorporated. It is an extension of the local optimizer SASS and improves a given solution along a search direction (no gradient information is used). Finally, a termination criteria has also been proposed, which stops the algorithm if during three consecutive iterations the changes experimented in the candidate Pareto-front are negligible (in terms of the objective function values). To know how far two sets are from each other, a modification of the well-known Hausdorff distance is proposed. In order to analyze the algorithm performance, it has been compared to the reference algorithms NSGA-II and SPEA2 and the state-of-the-art Keywords Nonlinear multi-objective optimization · evolutionary algorithm · quality indicators · computational study 1 Introduction

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Obtaining the efficient set of nonlinear biobjective optimization problems via interval branch-and-bound methods

Abstract: Nonlinear biobjective optimization, Efficient set, Outer approximation, Interval analysis, Branch-and-bound method, Discarding tests, Continuous location,

Cited by 41 publications

References 36 publications

A Triobjective Model for Locating a Public Semiobnoxious Facility in the Plane

A Triobjective Model for Locating a Public Semiobnoxious Facility in the Plane

Verified Methods for Computing Pareto Sets: General Algorithmic Analysis

FEMOEA: a fast and efficient multi-objective evolutionary algorithm

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