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2014 | Buch

Variable Ordering Structures Kapitel lesen Erstes Kapitel lesen

Variable Ordering Structures in Vector Optimization

verfasst von: Gabriele Eichfelder

Verlag: Springer Berlin Heidelberg

Buchreihe : Vector Optimization

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book provides an introduction to vector optimization with variable ordering structures, i.e., to optimization problems with a vector-valued objective function where the elements in the objective space are compared based on a variable ordering structure: instead of a partial ordering defined by a convex cone, we see a whole family of convex cones, one attached to each element of the objective space. The book starts by presenting several applications that have recently sparked new interest in these optimization problems, and goes on to discuss fundamentals and important results on a wide range of topics. The theory developed includes various optimality notions, linear and nonlinear scalarization functionals, optimality conditions of Fermat and Lagrange type, existence and duality results. The book closes with a collection of numerical approaches for solving these problems in practice.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Variable Ordering Structures
Abstract
In this chapter we recall basic concepts of partially ordered spaces and basic results on ordering cones and dual cones. We introduce variable ordering structures and examine their properties. We also focus on special ordering maps where the images are Bishop-Phelps cones and discuss several applications.
Gabriele Eichfelder
Chapter 2. Optimality Concepts and Their Characterization
Abstract
In this chapter we collect several optimality notions based on the two binary relations introduced in Chap.​ 1. We compare these new concepts with known concepts in partially ordered spaces and continue the chapter by providing first results on characterizations of the optimal elements.
Gabriele Eichfelder
Chapter 3. Cone-Valued Maps
Abstract
Variable ordering structures are defined by ordering maps which are cone-valued maps. For that reason we study in this chapter cone-valued maps. We examine classical properties, formerly introduced for arbitrary set-valued maps, like convexity, cone-convexity, linearity or monotonicity. It turns out that some of these properties like convexity directly imply that the cone-valued map is constant. In case of non-appropriateness of the classical notions we propose new concepts
Gabriele Eichfelder
Chapter 4. Linear Scalarizations
Abstract
In this chapter linear scalarization functionals are studied and characterization results are provided. With these functionals at hand a vector optimization problem can be replaced by a scalar-valued optimization problem which allows for instance the formulation of optimality conditions or can be used as the base of numerical solution methods.
Gabriele Eichfelder
Chapter 5. Nonlinear Scalarizations
Abstract
As linear scalarization functionals are appropriate in case of convexity of the considered set only, we discuss nonlinear scalarization functionals in this chapter which allow a complete characterization of nondominated and minimal elements. We consider a modification of the so-called signed distance functional which was introduced by Hiriart-Urruty, and of a second functional called translative functional, which is known in the literature as Gerstewitz or Tammer-Weidner functional or Pascoletti-Serafini scalarization.
Gabriele Eichfelder
Chapter 6. Scalarizations for Variable Orderings Given by Bishop-Phelps Cones
Abstract
In this chapter we concentrate on variable ordering structures which are defined by ordering maps with images being Bishop-Phelps (BP) cones. This additional structure allows introducing a new scalarization functional which is also new in partially ordered spaces. Based on this functional we give complete characterizations of nondominated and minimal elements.
Gabriele Eichfelder
Chapter 7. Optimality Conditions for Vector Optimization Problems
Abstract
In this chapter we provide subdifferential information for the scalarization functionals introduced in Chap.​ 6. Based on that we are able to formulate necessary and sufficient optimality conditions of Fermat and Lagrange type for unconstrained and constrained vector optimization problems with (set-valued) objective maps mapping in a real linear space equipped with a variable ordering structure.
Gabriele Eichfelder
Chapter 8. Duality Results
Abstract
In this chapter we obtain duality results based on the linear scalarization functionals of Chap.​ 4 and the nonlinear scalarization functionals of Chap.​ 6. We also provide results concerning general duality for a primal and a dual set. It is interesting to see that the two optimality concepts, the nondominated and the minimal elements, which are in general not related in the sense that the one does not imply the other, are related by duality results.
Gabriele Eichfelder
Chapter 9. Numerical Methods
Abstract
In Chap.​ 9 we give a survey on numerical approaches for solving vector optimization problems with a variable ordering structure. We provide algorithms for solving finite discrete as well as continuous vector optimization problems.
Gabriele Eichfelder
Chapter 10. Outlook and Further Application Areas
Abstract
In this final chapter we give a short outlook on the appearance of variable ordering structures in vector variational inequalities, vector complementarity and equilibrium problems. We show that the theory of consumer demand in economics is also related to variable ordering structures. Finally, we discuss an application in the treatment planning in intensity-modulated radiation therapy.
Gabriele Eichfelder
Backmatter
Metadaten
Titel
Variable Ordering Structures in Vector Optimization
verfasst von
Gabriele Eichfelder
Copyright-Jahr
2014
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-54283-1
Print ISBN
978-3-642-54282-4
DOI
https://doi.org/10.1007/978-3-642-54283-1

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