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2011 | Buch

Vector Optimization

Theory, Applications, and Extensions

verfasst von: Johannes Jahn

Verlag: Springer Berlin Heidelberg

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SUCHEN

Über dieses Buch

Fundamentals and important results of vector optimization in a general setting are presented in this book. The theory developed includes scalarization, existence theorems, a generalized Lagrange multiplier rule and duality results. Applications to vector approximation, cooperative game theory and multiobjective optimization are described. The theory is extended to set optimization with particular emphasis on contingent epiderivatives, subgradients and optimality conditions. Background material of convex analysis being necessary is concisely summarized at the beginning.

This second edition contains new parts on the adaptive Eichfelder-Polak method, a concrete application to magnetic resonance systems in medical engineering and additional remarks on the contribution of F.Y. Edgeworth and V. Pareto. The bibliography is updated and includes more recent important publications.

Inhaltsverzeichnis

Frontmatter

Convex Analysis

Frontmatter
Chapter 1. Linear Spaces
Abstract
Although several results of the theory described in the second part of this book are also valid in a rather abstract setting we restrict our attention to real linear spaces. For convenience, we summarize in this chapter the well-known definitions of linear spaces and convex sets as well as the definition of (locally convex) topological linear spaces and we consider a partial ordering in such a linear setting. Finally, we investigate some special partially ordered linear spaces and list various known properties.
Johannes Jahn
Chapter 2. Maps on Linear Spaces
Abstract
In this chapter various important classes of maps are considered for which one obtains interesting results in vector optimization. We especially consider convex maps and their generalizations and also several types of differentials. It is the aim of this chapter to present a brief survey on these maps
Johannes Jahn
Chapter 3. Some Fundamental Theorems
Abstract
For the investigation of vector optimization problems we need various fundamental theorems of convex analysis which are presented in this section. First, we formulate Zorns lemma and the Hahn-Banach theorem and, as a consequence, we examine several types of separation theorems. Moreover, we discuss a James theorem on the characterization of weakly compact sets and we study two Krein-Rutman theorems on the extension of positive linear functionals and the existence of strictly positive linear functionals. Finally, we prove a Ljusternik theorem on certain tangent cones.
Johannes Jahn

Theory of Vector Optimization

Frontmatter
Chapter 4. Optimality Notions
Abstract
For the investigation of optimal elements of a nonempty subset of a partially ordered linear space one is mainly interested in minimal or maximal elements of this set. But in certain situations it also makes sense to study several variants of these concepts; for example, strongly minimal, properly minimal and weakly minimal elements (or strongly maximal, properly maximal and weakly maximal elements). It is the aim of this first chapter of the second part to present the definition of these optimality notions together with some examples.
Johannes Jahn
Chapter 5. Scalarization
Abstract
In general, scalarization means the replacement of a vector optimization problem by a suitable scalar optimization problem which is an optimization problem with a real-valued objective functional. It is a fundamental principle in vector optimization that optimal elements of a subset of a partially ordered linear space can be characterized as optimal solutions of certain scalar optimization problems. Since the scalar optimization theory is widely developed scalarization turns out sto be of great importance for the vector optimization theory.
Johannes Jahn
Chapter 6. Existence Theorems
Abstract
In this chapter we study assumptions which guarantee that at least one optimal element of a subset of a partially ordered linear space exists. These investigations will be done for the minimality, proper minimality and weak minimality notions. Strongly minimal elements are not considered because this optimality notion is too restrictive.
Johannes Jahn
Chapter 7. Generalized Lagrange Multiplier Rule
Abstract
In this chapter we present a generalization of the famous and wellknown Lagrange multiplier rule published in 1797. Originally, Lagrange formulated his rule for the optimization of a real-valued function under side-conditions in the form of equalities. In this context we investigate an abstract optimization problem with equality and inequality constraints. For this problem we derivea generalized multiplier rule as a necessary optimality condition and we show under which assumptions this multiplier rule is also sufficient for optimality. The results are also applied to multiobjective optimization problems.
Johannes Jahn
Chapter 8. Duality
Abstract
It is well-known from scalar optimization that, under appropriateassumptions, a maximization problem can beassociated to a given minimization problem so that both problems have the same optimal values. Such a duality between a minimization and a maximization problem can also be formulated in vector optimization. In the first section we present a general duality principle for vector optimization problems. The following sections are devoted to a duality theory for abstract optimization problems. A generalization of the duality results known from linear programming is also given.
Johannes Jahn

Mathematical Applications

Frontmatter
Chapter 9. Vector Approximation
Abstract
Vector approximation problems are abstract approximation problems where a vectorial norm is used instead of a usual (real-valued) norm. Many important results known from approximation theory can be extended to this vector-valued case. After a short introduction we examine the relationship between vector approximation and simultaneous approximation, and we present the so-called generalized Kolmogorov condition. Moreover, we consider nonlinear and linear Chebyshev vector approximation problems and we formulate a generalized alternation theorem for these problems.
Johannes Jahn
Chapter 10. Cooperative n Player Differential Games
Abstract
In contrast to the theory of cooperative games introduced by John von Neumann, this chapter is devoted to deterministic differential games with n players behaving exclusively cooperatively. Such games can be described as vector optimization problems. After some basic remarks on the cooperation concept we present necessary and sufficient conditions for optimal and weakly optimal controls concerning a system of ordinary differential equations. In the last section we discuss a special cooperative differential game with a linear differential equation in a Hilbert space.
Johannes Jahn

Engineering Applications

Frontmatter
Chapter 11. Theoretical Basics of Multiobjective Optimization
Abstract
This chapter introduces the basic concepts of multiobjective optimization. After the discussion of a simple example from structural engineering in the first section the definitions of several variants of the Edgeworth-Pareto optimality notion are presented: weakly, properly, strongly and essentially Edgeworth-Pareto optimal points. Relationships between these different concepts are investigated and simple examples illustrate these notions. The second section is devoted to the scalarization of multiobjective optimization problems. The weighted sum and the weighted Chebyshev norm approach are investigated in detail.
Johannes Jahn
Chapter 12. Numerical Methods
Abstract
During the past 40 years many methods have been developed for the numerical solution of multiobjective optimization problems. Many of these methods are only applicable to special problem classes. In this chapter we present only some few methods which can be applied to general multiobjective optimization problems. These are a method proposed by Polak and an extension given by Eichfelder, a method for discrete problems and in the class of interactive methods we present the STEM method and a method of reference point approximation. In principle, one can also use the scalarization results for the determination of an Edgeworth-Pareto optimal point. But then it remains an open question whether the determined Edgeworth-Pareto optimal point is the subjectively best for the decision maker.
Johannes Jahn
Chapter 13. Multiobjective Design Problems
Abstract
Multiobjective optimization problems turn up in almost all fields of engineering. The application areas range from designs of electrical switching circuits, machine parts, airplanes and weight-bearing structures (bridges, pylons etc.) to planning and controlling of watersupply systems.
Johannes Jahn

Extensions to Set Optimization

Frontmatter
Chapter 14. Basic Concepts and Results of Set Optimization
Abstract
In this chapter we consider vector optimization problems with a setvalued objective map which has to be minimized or maximized. For these set optimization problems we present basic concepts and first results.
Johannes Jahn
Chapter 15. Contingent Epiderivatives
Abstract
For the formulation of optimality conditions one needs an appropriate differentiability concept for set-valued maps. In this chapter we present the notion of contingent epiderivatives and generalized contingent epiderivatives. We show properties of these contingent epiderivatives and discuss the special case of real-valued functions.
Johannes Jahn
Chapter 16. Subdifferential
Abstract
There are different possibilities to introduce subgradients of set-valued maps. One possible approach is a generalization of the standard definition known from convex analysis (see also Definition 2.21). Another approach is based on a characterization of the subdifferential using directional derivatives (e.g., see [164, Lemma 3.25]). Instead of the directional derivative we now use the contingent epiderivative. Both approaches are presented in this chapter.
Johannes Jahn
Chapter 17. Optimality Conditions
Abstract
Based on the concepts introduced in the preceding chapters we now present optimality conditions for set optimization problems. These conditions are discussed using contingent epiderivatives, subgradients and weak subgradients. The main section of this chapter is devoted to a generalization of the Lagrange multiplier rule. We present this multiplier rule as a necessary optimality condition. Assumptions ensuring that this multiplier rule is a sufficient optimality condition are also given.
Johannes Jahn
Backmatter
Metadaten
Titel
Vector Optimization
verfasst von
Johannes Jahn
Copyright-Jahr
2011
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-17005-8
Print ISBN
978-3-642-17004-1
DOI
https://doi.org/10.1007/978-3-642-17005-8