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Mathematics
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Recent Trends in Multiobjective Optimization and Optimal Control
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Recent Trends in Multiobjective Optimization and Optimal Control

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (29 February 2020) | Viewed by 14355

Special Issue Editors

Prof. Dr. Michael Dellnitz

E-Mail Website
Guest Editor
Chair of Applied Mathematics, University of Paderborn, 33098 Paderborn, Germany
Interests: dynamical systems; multiobjective optimization
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Sina Ober-Bloebaum

E-Mail Website
Guest Editor
Department of Engineering Science, University of Oxford, Oxford, UK
Interests: multiobjective optimal control; geometric integration; computational mechanics
Dr. Sebastian Peitz

E-Mail Website
Guest Editor
Data Science for Engineering, Department of Computer Science, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany
Interests: multiobjective optimization; optimal control; fluid dynamics and flow control; dynamical systems; data-driven modeling; model order reduction
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In many applications in the natural sciences, in engineering or in industry one has to optimize several objectives at the same time. A topical paradigm is the simultaneous optimization of safety, energy efficiency and comfort in the area of autonomous driving. In mathematical terms this leads to multiobjective optimization problems whose solution—the “optimal compromise” between the different objectives—is provided by the set of so-called Pareto optimal points.

Due to the increasing complexity of current technical innovations, interest in the field of multiobjective optimization has recently increased significantly. The purpose of this Special Issue is to highlight recent trends and significant advances in this area. Of interest in this context are topics including but not limited to:

  • multiobjective optimal/model predictive/feedback control;
  • multiobjective optimization with PDE constraints;
  • many-objective optimization (i.e. the number of objectives is larger than 3);
  • multiobjective optimization based on data, or in combination with mathematical models (e.g. for the problems mentioned above).

Approaches to the solution of these problems are e.g. provided by classical mathematical optimization methods, modern control strategies, techniques from model order reduction, evolutionary algorithms, robust and data-driven optimization or from machine learning. Each submission should either directly address or elaborate on the relevance for an application. However, the wide spectrum of applications is not limited and it could range from real-time applications in autonomous driving to control problems in fluid mechanics.

Prof. Dr. Michael Dellnitz
Prof. Dr. Sina Ober-Bloebaum
Dr. Sebastian Peitz
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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
  • multiobjective optimal control
  • data based control
  • reduced order modelling
  • surrogate modelling

Published Papers (5 papers)

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Research

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19 pages, 1057 KiB  
Article
Multiobjective Model Predictive Control of a Parabolic Advection-Diffusion-Reaction Equation
by Stefan Banholzer, Giulia Fabrini, Lars Grüne and Stefan Volkwein
Mathematics 2020, 8(5), 777; https://doi.org/10.3390/math8050777 - 12 May 2020
Cited by 8 | Viewed by 1649
Abstract
In the present paper, a multiobjective optimal control problem governed by a linear parabolic advection-diffusion-reaction equation is considered. The optimal controls are computed by applying model predictive control (MPC), which is a method for controlling dynamical systems over long or infinite time horizons [...] Read more.
In the present paper, a multiobjective optimal control problem governed by a linear parabolic advection-diffusion-reaction equation is considered. The optimal controls are computed by applying model predictive control (MPC), which is a method for controlling dynamical systems over long or infinite time horizons by successively computing optimal controls over a moving finite time horizon. Numerical experiments illustrate that the proposed solution approach can be successfully applied although some of the assumptions which are necessary to conduct the theoretical analysis cannot be guaranteed for the studied tests. Full article
(This article belongs to the Special Issue Recent Trends in Multiobjective Optimization and Optimal Control)
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29 pages, 2444 KiB  
Article
Multiobjective Mixed Integer Nonlinear Model to Plan the Schedule for the Final Disposal of the Spent Nuclear Fuel in Finland
by Outi Montonen, Ville-Pekka Eronen, Timo Ranta, Jani A. S. Huttunen and Marko M. Mäkelä
Mathematics 2020, 8(4), 528; https://doi.org/10.3390/math8040528 - 03 Apr 2020
Cited by 1 | Viewed by 1859
Abstract
The safe disposal of the spent nuclear fuel is the important part of the nuclear power production. In this paper, we model the geological disposal in Finland covering objectives related to the interim storage, the encapsulation facility, the disposal facility, and the costs. [...] Read more.
The safe disposal of the spent nuclear fuel is the important part of the nuclear power production. In this paper, we model the geological disposal in Finland covering objectives related to the interim storage, the encapsulation facility, the disposal facility, and the costs. A notable fact is that all the fuel types used in Finland are taken into account. The resulting optimization model is of a multiobjective nonlinear mixed integer type having eight objectives. The model is solved with the interactive method utilizing the special type of the achievement scalarizing functions. From this, we obtain a disposal schedule giving amounts of canisters to encapsulate in each time period. The results obtained are analyzed from the practical point of view. Full article
(This article belongs to the Special Issue Recent Trends in Multiobjective Optimization and Optimal Control)
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25 pages, 1764 KiB  
Article
A New Hybrid Evolutionary Algorithm for the Treatment of Equality Constrained MOPs
by Oliver Cuate, Antonin Ponsich, Lourdes Uribe, Saúl Zapotecas-Martínez, Adriana Lara and Oliver Schütze
Mathematics 2020, 8(1), 7; https://doi.org/10.3390/math8010007 - 18 Dec 2019
Cited by 18 | Viewed by 2804
Abstract
Multi-objective evolutionary algorithms are widely used by researchers and practitioners to solve multi-objective optimization problems (MOPs), since they require minimal assumptions and are capable of computing a finite size approximation of the entire solution set in one run of the algorithm. So far, [...] Read more.
Multi-objective evolutionary algorithms are widely used by researchers and practitioners to solve multi-objective optimization problems (MOPs), since they require minimal assumptions and are capable of computing a finite size approximation of the entire solution set in one run of the algorithm. So far, however, the adequate treatment of equality constraints has played a minor role. Equality constraints are particular since they typically reduce the dimension of the search space, which causes problems for stochastic search algorithms such as evolutionary strategies. In this paper, we show that multi-objective evolutionary algorithms hybridized with continuation-like techniques lead to fast and reliable numerical solvers. For this, we first propose three new problems with different characteristics that are indeed hard to solve by evolutionary algorithms. Next, we develop a variant of NSGA-II with a continuation method. We present numerical results on several equality-constrained MOPs to show that the resulting method is highly competitive to state-of-the-art evolutionary algorithms. Full article
(This article belongs to the Special Issue Recent Trends in Multiobjective Optimization and Optimal Control)
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24 pages, 1246 KiB  
Article
An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-Spline Data Approximation and Trajectory Optimization
by Jens Jauch, Felix Bleimund, Michael Frey and Frank Gauterin
Mathematics 2019, 7(4), 355; https://doi.org/10.3390/math7040355 - 16 Apr 2019
Cited by 1 | Viewed by 3375
Abstract
The B-spline function representation is commonly used for data approximation and trajectory definition, but filter-based methods for nonlinear weighted least squares (NWLS) approximation are restricted to a bounded definition range. We present an algorithm termed nonlinear recursive B-spline approximation (NRBA) for an iterative [...] Read more.
The B-spline function representation is commonly used for data approximation and trajectory definition, but filter-based methods for nonlinear weighted least squares (NWLS) approximation are restricted to a bounded definition range. We present an algorithm termed nonlinear recursive B-spline approximation (NRBA) for an iterative NWLS approximation of an unbounded set of data points by a B-spline function. NRBA is based on a marginalized particle filter (MPF), in which a Kalman filter (KF) solves the linear subproblem optimally while a particle filter (PF) deals with nonlinear approximation goals. NRBA can adjust the bounded definition range of the approximating B-spline function during run-time such that, regardless of the initially chosen definition range, all data points can be processed. In numerical experiments, NRBA achieves approximation results close to those of the Levenberg–Marquardt algorithm. An NWLS approximation problem is a nonlinear optimization problem. The direct trajectory optimization approach also leads to a nonlinear problem. The computational effort of most solution methods grows exponentially with the trajectory length. We demonstrate how NRBA can be applied for a multiobjective trajectory optimization for a battery electric vehicle in order to determine an energy-efficient velocity trajectory. With NRBA, the effort increases only linearly with the processed data points and the trajectory length. Full article
(This article belongs to the Special Issue Recent Trends in Multiobjective Optimization and Optimal Control)
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Review

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35 pages, 820 KiB  
Review
The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review
by Johan M. Bogoya, Andrés Vargas and Oliver Schütze
Mathematics 2019, 7(10), 894; https://doi.org/10.3390/math7100894 - 24 Sep 2019
Cited by 23 | Viewed by 3831
Abstract
A brief but comprehensive review of the averaged Hausdorff distances that have recently been introduced as quality indicators in multi-objective optimization problems (MOPs) is presented. First, we introduce all the necessary preliminaries, definitions, and known properties of these distances in order to provide [...] Read more.
A brief but comprehensive review of the averaged Hausdorff distances that have recently been introduced as quality indicators in multi-objective optimization problems (MOPs) is presented. First, we introduce all the necessary preliminaries, definitions, and known properties of these distances in order to provide a stat-of-the-art overview of their behavior from a theoretical point of view. The presentation treats separately the definitions of the ( p , q ) -distances GD p , q , IGD p , q , and Δ p , q for finite sets and their generalization for arbitrary measurable sets that covers as an important example the case of continuous sets. Among the presented results, we highlight the rigorous consideration of metric properties of these definitions, including a proof of the triangle inequality for distances between disjoint subsets when p , q ⩾ 1 , and the study of the behavior of associated indicators with respect to the notion of compliance to Pareto optimality. Illustration of these results in particular situations are also provided. Finally, we discuss a collection of examples and numerical results obtained for the discrete and continuous incarnations of these distances that allow for an evaluation of their usefulness in concrete situations and for some interesting conclusions at the end, justifying their use and further study. Full article
(This article belongs to the Special Issue Recent Trends in Multiobjective Optimization and Optimal Control)
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