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Symmetry
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Symmetry Techniques for Multiobjective Optimization in Finite and Infinite Dimensions
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Symmetry Techniques for Multiobjective Optimization in Finite and Infinite Dimensions

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 11786

Special Issue Editors

Prof. Dr. Francisco Javier Garcia-Pacheco

E-Mail Website
Guest Editor
Department of Mathematics, College of Engineering, Universidad de Cádiz, 11510 Puerto Real, Spain
Interests: functional analysis; algebra; geometry; topology
Special Issues, Collections and Topics in MDPI journals
Prof. Marina Murillo Arcila

E-Mail Website
Guest Editor
Department of Mathematics, Sciences Faculty, University of Cadiz, E-11519 Puerto Real, Spain
Interests: functional analysis; algebra; geometry; topology

Special Issue Information

Dear Colleagues,

A large number of problems in Bioengineering, Physics and Statistics can be modeled as multiobjective optimization problems. This kind of optimization problems also arise in many mathematical fields, such as the modeling of dynamical systems or networks. Sometimes, they are very hard to fully solve, that is, a full solution that optimizes all the objective functions at once might not actually exist. This is why Pareto optimality comes into play. However, it may even be hard to find all the Pareto optimal solutions of a multiobjective optimization problem. Therefore, it is sometimes necessary to reformulate the multiobjective optimization problem to obtain a simpler optimization problem that preserves the Pareto optimal solutions.

This Special Issue is devoted to collecting all new original results in this trend together with applications to real life situations that show the validity of the theoretical results. A functional analysis approach to multiobjective optimization problems is very welcome in this Special Issue because this kind of approach also works in infinite dimensions, whereas multiobjective optimization problems are typically approached from finite-dimensional settings.

Prof. Francisco Javier Garcia-Pacheco
Prof. Marina Murillo Arcila
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Multiobjective optimization
  • Pareto optimality
  • Supporting vector
  • Tychonov regularization
  • Minimum-norm problems
  • Convex optimization
  • Operator norms
  • Minimax optimization
  • Fixed-point techniques
  • Dynamics
  • Networks

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Published Papers (6 papers)

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Research

12 pages, 266 KiB  
Article
Lie Symmetries and Conservation Laws for the Viscous Cahn-Hilliard Equation
by Almudena P. Márquez, Elena Recio and María L. Gandarias
Symmetry 2022, 14(5), 861; https://doi.org/10.3390/sym14050861 - 22 Apr 2022
Cited by 2 | Viewed by 1726
Abstract
In this paper, we study a viscous Cahn–Hilliard equation from the point of view of Lie symmetries in partial differential equations. The analysis of this equation is motivated by its applications since it serves as a model for many problems in physical chemistry, [...] Read more.
In this paper, we study a viscous Cahn–Hilliard equation from the point of view of Lie symmetries in partial differential equations. The analysis of this equation is motivated by its applications since it serves as a model for many problems in physical chemistry, developmental biology, and population movement. Firstly, a classification of the Lie symmetries admitted by the equation is presented. In addition, the symmetry transformation groups are calculated. Afterwards, the partial differential equation is transformed into ordinary differential equations through symmetry reductions. Secondly, all low-order local conservation laws are obtained by using the multiplier method. Furthermore, we use these conservation laws to determine their associated potential systems and we use them to investigate nonlocal symmetries and nonlocal conservation laws. Finally, we apply the multi-reduction method to reduce the equation and find a soliton solution. Full article
(This article belongs to the Special Issue Symmetry Techniques for Multiobjective Optimization in Finite and Infinite Dimensions)
18 pages, 303 KiB  
Article
Using Shapley Values and Genetic Algorithms to Solve Multiobjective Optimization Problems
by Hsien-Chung Wu
Symmetry 2021, 13(11), 2021; https://doi.org/10.3390/sym13112021 - 25 Oct 2021
Cited by 5 | Viewed by 2139
Abstract
This paper proposes a new methodology to solve multiobjective optimization problems by invoking genetic algorithms and the concept of the Shapley values of cooperative games. It is well known that the Pareto-optimal solutions of multiobjective optimization problems can be obtained by solving the [...] Read more.
This paper proposes a new methodology to solve multiobjective optimization problems by invoking genetic algorithms and the concept of the Shapley values of cooperative games. It is well known that the Pareto-optimal solutions of multiobjective optimization problems can be obtained by solving the corresponding weighting problems that are formulated by assigning some suitable weights to the objective functions. In this paper, we formulated a cooperative game from the original multiobjective optimization problem by regarding the objective functions as the corresponding players. The payoff function of this formulated cooperative game involves the symmetric concept, which means that the payoff function only depends on the number of players in a coalition and is independent of the role of players in this coalition. In this case, we can reasonably set up the weights as the corresponding Shapley values of this formulated cooperative game. Under these settings, we can obtain the so-called Shapley–Pareto-optimal solution. In order to choose the best Shapley–Pareto-optimal solution, we used genetic algorithms by setting a reasonable fitness function. Full article
(This article belongs to the Special Issue Symmetry Techniques for Multiobjective Optimization in Finite and Infinite Dimensions)
10 pages, 279 KiB  
Article
An Advised Indirect-Utility Ranking of Opportunity Sets
by María Dolores García-Sanz and José Carlos R. Alcantud
Symmetry 2021, 13(8), 1404; https://doi.org/10.3390/sym13081404 - 2 Aug 2021
Viewed by 1439
Abstract
There is a substantial strand of literature about ranking the subsets of a set of alternatives, usually referred to as opportunity sets. We investigate a model that is dependent on the preference of a grand set of alternatives. In this framework, the indirect-utility [...] Read more.
There is a substantial strand of literature about ranking the subsets of a set of alternatives, usually referred to as opportunity sets. We investigate a model that is dependent on the preference of a grand set of alternatives. In this framework, the indirect-utility criterion ranks the opportunity sets by the following rule: a subset A is weakly preferred to another subset B if and only if A contains at least one preference maximizing element from A∪B. This criterion leads to the indifference of each subset of alternatives to a singleton; symmetry appears at this stage, as the property holds true for any one of the maximizers in A. Conversely, suppose that a ranking of opportunity sets satisfies the property that each opportunity set is indifferent to a singleton contained within it. Then, we prove that such a ranking must use a generalized form of the indirect-utility criterion, where maximization is applied to a selection of the alternatives. Altogether, these results produce a characterization of the advised indirect-utility criterion for ranking opportunity sets. Full article
(This article belongs to the Special Issue Symmetry Techniques for Multiobjective Optimization in Finite and Infinite Dimensions)
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9 pages, 272 KiB  
Article
A Note on Decomposable and Reducible Integer Matrices
by Carlos Marijuán, Ignacio Ojeda and Alberto Vigneron-Tenorio
Symmetry 2021, 13(7), 1125; https://doi.org/10.3390/sym13071125 - 24 Jun 2021
Viewed by 1589
Abstract
We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix, we associate a symmetric integer matrix whose reducibility can be efficiently determined by elementary linear algebra techniques, and [...] Read more.
We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix, we associate a symmetric integer matrix whose reducibility can be efficiently determined by elementary linear algebra techniques, and which completely determines the decomposability of the first one. Full article
(This article belongs to the Special Issue Symmetry Techniques for Multiobjective Optimization in Finite and Infinite Dimensions)
6 pages, 227 KiB  
Article
Counting the Ideals with a Given Genus of a Numerical Semigroup with Multiplicity Two
by M. A. Moreno-Frías and José Carlos Rosales
Symmetry 2021, 13(5), 794; https://doi.org/10.3390/sym13050794 - 3 May 2021
Cited by 2 | Viewed by 1598
Abstract
Let S and T be two numerical semigroups. We say that T is an I(S)-semigroup if T∖{0} is an ideal of S. Given k a positive integer, we denote by [...] Read more.
Let S and T be two numerical semigroups. We say that T is an I(S)-semigroup if T∖{0} is an ideal of S. Given k a positive integer, we denote by Δ(k) the symmetric numerical semigroup generated by {2,2k+1}. In this paper we present a formula which calculates the number of I(S)-semigroups with genus g(Δ(k))+h for some nonnegative integer h and which we will denote by i(Δ(k),h). As a consequence, we obtain that the sequence {i(Δ(k),h)}h∈N is never decreasing. Besides, it becomes stationary from a certain term. Full article
(This article belongs to the Special Issue Symmetry Techniques for Multiobjective Optimization in Finite and Infinite Dimensions)
17 pages, 750 KiB  
Article
Pareto Optimality for Multioptimization of Continuous Linear Operators
by Clemente Cobos-Sánchez, José Antonio Vilchez-Membrilla, Almudena Campos-Jiménez and Francisco Javier García-Pacheco
Symmetry 2021, 13(4), 661; https://doi.org/10.3390/sym13040661 - 12 Apr 2021
Cited by 6 | Viewed by 2000
Abstract
This manuscript determines the set of Pareto optimal solutions of certain multiobjective-optimization problems involving continuous linear operators defined on Banach spaces and Hilbert spaces. These multioptimization problems typically arise in engineering. In order to accomplish our goals, we first characterize, in an abstract [...] Read more.
This manuscript determines the set of Pareto optimal solutions of certain multiobjective-optimization problems involving continuous linear operators defined on Banach spaces and Hilbert spaces. These multioptimization problems typically arise in engineering. In order to accomplish our goals, we first characterize, in an abstract setting, the set of Pareto optimal solutions of any multiobjective optimization problem. We then provide sufficient topological conditions to ensure the existence of Pareto optimal solutions. Next, we determine the Pareto optimal solutions of convex max–min problems involving continuous linear operators defined on Banach spaces. We prove that the set of Pareto optimal solutions of a convex max–min of form max∥T(x)∥, min∥x∥ coincides with the set of multiples of supporting vectors of T. Lastly, we apply this result to convex max–min problems in the Hilbert space setting, which also applies to convex max–min problems that arise in the design of truly optimal coils in engineering. Full article
(This article belongs to the Special Issue Symmetry Techniques for Multiobjective Optimization in Finite and Infinite Dimensions)
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