Multi-Objective Optimization in Theory and Practice I: Classical Methods

by

André A. Keller

DOI: 10.2174/97816810856851170101
eISBN: 978-1-68108-568-5, 2017
ISBN: 978-1-68108-569-2



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Multi-Objective Optimization in Theory and Practice is a traditional two-part approach to solving multi-objective opt...[view complete introduction]
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Table of Contents

Preface

- Pp. i-ii (2)
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Acknowledgements

- Pp. iii
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Design of Multi-Objective Optimization Problems

- Pp. 1-16 (16)
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Extended Multi-Objective Optimization Problems

- Pp. 17-45 (29)
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Pareto Optimality

- Pp. 46-66 (21)
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Founding Multi-Optimization Techniques

- Pp. 67-108 (42)
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Preference-Based Methods of MOO Problems

- Pp. 109-137 (29)
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Mixed-Integer Nonlinear Programming

- Pp. 138-172 (35)
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Nested Optimization of Hierarchical Systems

- Pp. 173-203 (31)
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Fuzzy Logic Programming

- Pp. 204-237 (34)
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Noncooperative Games in a Fuzzy Environment

- Pp. 238-260 (23)
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List of Abbreviations

- Pp. 261-262 (2)
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List of Symbols

- Pp. 263-269 (7)
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Author Index

- Pp. 270-273 (4)
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Subject Index

- Pp. 274-282 (9)
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Preface

The mathematical foundations of programming relate to two significant contributions, namely the Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern in 1944, and the simplex method discovered by George B. Dantzig in 1947. Paradoxically, the origin of multi-objective optimization (MOO) goes back to earlier works by Francis Y. Edgeworth in 1881, and Vilfredo Pareto in 1906. The two leading economists developed the theory of indifference curves and defined the basic concept of optimality for problems where multiple objectives are to be optimized. In such problems, the objectives are often conflicting. The goal is to find trade-off solutions that represent best compromises among the objectives. Classical methods for solving MOO problems include the Zeleny’s simplex algorithm, the weighting methods, simplex method, and the preferencebased techniques including the value function method, goal programming, and the generalized center method. The development of multi-objective approach accelerated in the mid-1980s with the help of evolutionary algorithms for solving real-world multi-objective problems. A recent review by Fister et al. (2013) collected 74 nature-inspired metaheuristic algorithms. This project consists of two eBooks on “Multi-objective optimization in theory and practice: I: Classical methods and II: Evolutionary algorithms”. This project is an attempt to handle most significant aspects of the real-life complexities in decision-making. Various difficulties arise from non-convexities, nonlinearities, and non-differentiability of objective functions and side-constraints, multiple conflicting goals, a multiplicity of global and local solutions, uncertainties due to inaccurate data, and so forth. This project originated in lectures given the author in Operations Research and econometrics at the University of Paris. More recently, the author participated as a plenary speaker in several international conferences notably on game theory, and evolutionary optimization. This project reviews and evaluates multi-objective programming models using several software packages. The eBook I on classical methods for solving MOO problems includes nine Chapters. The first three Chapters focus on the design of current MOO problems (Chapter 1), the complexity of MOO problems with nonlinearities and uncertainties (Chapter 2), and the theory of Pareto optimality (Chapter 3). The next two Chapters introduce the founding solving methods (Chapter 4) and the preference-based methods (Chapter 5). The last four Chapters deal with the special structures of MOO problems, such as the mixedinteger programming (Chapter 6), hierarchical optimization (Chapter 7), fuzzy logic programming (Chapter 8), and bimatrix games (Chapter 9). This project is typically user-oriented with theoretical and practical aspects. This project includes detailed examples, illustrative figures, test functions, and small-size applications from the literature. The calculations are performed using, in particular, the commercial Mathematica® software, and implementing other free software packages1.

Prof. André A. Keller2
Center for Research in Computer Science
Signal and Automatic Control of Lille
University of Lille – CNRS
France



1 The reader's interest is thought in this book. The choice was to retain a leading commercial software and other free software packages easy to implement. Mathematica®'s interest at this point is his analytical language close to the current math expression. The choice of Mathematica®Version 7 (the current version is the 11th) reflects the experience of the author on this package since its creation early 1990. Another interest of Mathematica® is to have several extra packages in areas such as discrete mathematics (i.e., combinatorics and graph theory), fuzzy logic, and game theory. The use of another software than Mathematica® is more expanded in volume 2 of this eBook. I said that free software SciLab 5.5.2 is a very close commercial MatLab software.

2 Address correspondance to Prof. André A. Keller: Associate Researcher, Equipe Systèmes Multi-Agents et Comportement (SMAC), Centre de Recherche en Informatique, Signal et Automatique de Lille (CRISTAL), Université de Lille, Bâtiment. M3 Extension, 59655 Villeneuve d’Ascq, France. E-mail : andre.keller@univ-lille1.fr.

List of Contributors

Author(s):
André A. Keller
Center for Research in Computer Science
Signal and Automatic Control of Lille
University of Lille – CNRS
France




Reviews

Review 1

Multi-Objective Optimization in Theory and Practice I: Classical Methods

"The structure of the book is nicely arranged and representation of the given concepts is appropriate and is easy to understand for readers" - Ichiro Nizhizaki, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, Japan


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