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1998 | Buch

Concepts Kapitel lesen Erstes Kapitel lesen

Nonlinear Multiobjective Optimization

verfasst von: Kaisa Miettinen, PhD

Verlag: Springer US

Buchreihe : International Series in Operations Research & Management Science

Enthalten in: Professional Book Archive

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Über dieses Buch

Problems with multiple objectives and criteria are generally known as multiple criteria optimization or multiple criteria decision-making (MCDM) problems. So far, these types of problems have typically been modelled and solved by means of linear programming. However, many real-life phenomena are of a nonlinear nature, which is why we need tools for nonlinear programming capable of handling several conflicting or incommensurable objectives. In this case, methods of traditional single objective optimization and linear programming are not enough; we need new ways of thinking, new concepts, and new methods - nonlinear multiobjective optimization.
Nonlinear Multiobjective Optimization provides an extensive, up-to-date, self-contained and consistent survey, review of the literature and of the state of the art on nonlinear (deterministic) multiobjective optimization, its methods, its theory and its background. The amount of literature on multiobjective optimization is immense. The treatment in this book is based on approximately 1500 publications in English printed mainly after the year 1980.
Problems related to real-life applications often contain irregularities and nonsmoothnesses. The treatment of nondifferentiable multiobjective optimization in the literature is rather rare. For this reason, this book contains material about the possibilities, background, theory and methods of nondifferentiable multiobjective optimization as well.
This book is intended for both researchers and students in the areas of (applied) mathematics, engineering, economics, operations research and management science; it is meant for both professionals and practitioners in many different fields of application. The intention has been to provide a consistent summary that may help in selecting an appropriate method for the problem to be solved. It is hoped the extensive bibliography will be of value to researchers.

Inhaltsverzeichnis

Frontmatter

Terminology And Theory

Frontmatter
2. Concepts
Abstract
This chapter introduces the basic concepts of (nonlinear) multiobjective optimization and the notations used in the continuation.
Kaisa Miettinen
3. Theoretical Background
Abstract
We present a set of optimality conditions for multiobjective optimization problems. Because the conditions are different for differentiable and nondiffer-entiable problem, they are handled separately.
Kaisa Miettinen

Methods

Frontmatter
2. No-Preference Methods
Abstract
In no-preference methods, where the opinions of the decision maker are not taken into consideration, the multiobjective optimization problem is solved using some relatively simple method and the solution obtained is presented to the decision maker. The decision maker may either accept or reject the solution. It seems quite unlikely that the solution best satisfying the decision maker could be found with these methods. That is why no-preference methods are suitable for situations where the decision maker does not have any special expectations of the solution and (s)he is satisfied simply with some optimal solution. The working order here is: 1) analyst, 2) none.
Kaisa Miettinen
3. A Posteriori Methods
Abstract
A posteriori methods could also be called methods for generating Pareto optimal solutions. After the Pareto optimal set (or a part of it) has been generated, it is presented to the decision maker, who selects the most preferred among the alternatives. The inconveniences here are that the generation process is usually computationally expensive and sometimes in part, at least, difficult. On the other hand, it is hard for the decision maker to select from a large set of alternatives. One more important question is how to present or display the alternatives to the decision maker in an effective way. The working order in these methods is: 1) analyst, 2) decision maker.
Kaisa Miettinen
4. A Priori Methods
Abstract
In the case of a priori methods, the decision maker must specify her or his preferences, hopes and opinions before the solution process. The difficulty is that the decision maker does not necessarily know beforehand what it is possible to attain in the problem and how realistic her or his expectations are. The working order in these methods is: 1) decision maker, 2) analyst.
Kaisa Miettinen
5. Interactive Methods
Abstract
The class of interactive methods is the most developed of the four classes of methods presented here. The interest devoted to this class can be explained by the fact that assuming the decision maker has enough time and capabilities for co-operation, interactive methods can be presumed to produce the most satisfactory results. Many of the weak points of the methods in the other three classes are overcome. Namely, only part of the Pareto optimal points has to be generated and evaluated, and the decision maker can specify and correct her or his preferences and selections as the solution process continues and (s)he gets to know the problem and its potentialities better. This also means that the decision maker does not have to know any global preference structure. In addition, the decision maker can be assumed to have more confidence in the final solution since (s)he is involved throughout the solution process.
Kaisa Miettinen

Related Issues

Frontmatter
1. Comparing Methods
Abstract
As has been stressed many times thus far, a large variety of methods exists for multiobjective optimization problems and none of them can be claimed to be superior to the others in every aspect. Selecting a multiobjective optimization method is a problem with multiple objectives itself. Thus some matters of comparison and selection between the methods are worth considering.
Kaisa Miettinen
2. Software
Abstract
The development of computers and the improvement in the speed, storage capacities and flexibility of computing facilities have made it possible to produce more sophisticated and demanding software for solving multiobjec-tive optimization problems. Efficient computers enable, for example, the implementation of interactive algorithms, since they can produce sufficiently fast responses for the decision maker without the user getting frustrated waiting.
Kaisa Miettinen
3. Graphical Illustration
Abstract
Graphical illustration plays an essential role when designing modern software user interfaces. Graphics may be used to describe the problem, to assist the decision maker in specifying values for problem parameters or to illustrate the contents and the meaning of questions posed by the algorithms. In such realizations, the upper limit lies in one’s imagination.
Kaisa Miettinen
4. Future Directions
Abstract
In this chapter, we outline some challenging topics for the future development of multiobjective optimization, mainly from a mathematical point of view. In addition, we give examples of promising ideas for research where the first steps have been taken but further work is needed. All the issues mentioned and many others merit further research and examination.
Kaisa Miettinen
5. Epilogue
Abstract
We have presented a self-contained survey of the state of the art of nonlinear multiobjective optimization together with a great number of further references. After treating several important concepts and their relations, we have considered some theoretical results and connections.
Kaisa Miettinen
Backmatter
Metadaten
Titel
Nonlinear Multiobjective Optimization
verfasst von
Kaisa Miettinen, PhD
Copyright-Jahr
1998
Verlag
Springer US
Electronic ISBN
978-1-4615-5563-6
Print ISBN
978-1-4613-7544-9
DOI
https://doi.org/10.1007/978-1-4615-5563-6