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2018 | Buch

Kapitel lesen Erstes Kapitel lesen

Vector Variational Inequalities and Vector Optimization

Theory and Applications

verfasst von: Prof. Dr. Qamrul Hasan Ansari, Dr. Elisabeth Köbis, Prof. Dr. Jen-Chih Yao

Verlag: Springer International Publishing

Buchreihe : Vector Optimization

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

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Über dieses Buch

This book presents the mathematical theory of vector variational inequalities and their relations with vector optimization problems. It is the first-ever book to introduce well-posedness and sensitivity analysis for vector equilibrium problems. The first chapter provides basic notations and results from the areas of convex analysis, functional analysis, set-valued analysis and fixed-point theory for set-valued maps, as well as a brief introduction to variational inequalities and equilibrium problems. Chapter 2 presents an overview of analysis over cones, including continuity and convexity of vector-valued functions. The book then shifts its focus to solution concepts and classical methods in vector optimization. It describes the formulation of vector variational inequalities and their applications to vector optimization, followed by separate chapters on linear scalarization, nonsmooth and generalized vector variational inequalities. Lastly, the book introduces readers to vector equilibrium problems and generalized vector equilibrium problems. Written in an illustrative and reader-friendly way, the book offers a valuable resource for all researchers whose work involves optimization and vector optimization.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract

This chapter deals with basic definitions from convex analysis and nonlinear analysis, such as convex sets and cones, convex functions and their properties, generalized derivatives, and continuity for set-valued maps. We also gather some known results from fixed point theory for set-valued maps, namely, Nadler’s fixed point theorem, Fan-KKM lemma and its generalizations, Fan section lemma and its generalizations, Browder fixed point theorem and its generalizations, maximal element theorems and Kakutani fixed point theorem. A brief introduction of scalar variational inequalities, nonsmooth variational inequalities, generalized variational inequalities and equilibrium problems is given.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Chapter 2. Analysis over Cones

Abstract

This chapter presents an overview of analysis over cones, including the definition and study of binary relations, orders, and orders that are induces by convex cones. Cone topological concepts are investigated. Moreover, continuity and convexity notions of vector-valued functions are discussed. As the most useful and common practice to solve a vector optimization problem is to convert it into a scalar optimization problem, a nonlinear scalarizing functional is introduced. It is shown that such a functional is an important tool for separating nonconvex sets and several properties of this functional are studied. These basic tools from convex analysis and nonlinear analysis provide the reader with the necessary knowledge needed for studying vector optimization problems and vector variational inequalities.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Chapter 3. Solution Concepts in Vector Optimization

Abstract

Many applications require the optimization of multiple conflicting goals at the same time. Such a problem can be modeled as a vector optimization problem. Vector optimization deals with the problem of finding efficient elements of a vector-valued function. In that sense, vector optimization generalizes the concept of scalar optimization. In scalar optimization, there is only one concept for efficiency which characterizes efficient elements, namely the solution which generates the smallest function value. But, due to the lack of a total order in general spaces, order relations that are defined within the optimality concept need to be chosen. In this chapter, we discuss several solution concepts for a vector optimization problem. In particular, solution concepts for vector optimization problem equipped with a variable domination structure are studied. Moreover, we present some existence results for solutions of vector optimization problems.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Chapter 4. Classical Methods in Vector Optimization

Abstract

In this chapter, we investigate solution procedures to obtain efficient solutions of a vector optimization problem.

The results that we are recalling in this chapter are standard and can be found in several books on vector optimization, for example in Chankong and Haimes (Multiobjective decision making: theory and methodology. Elsevier Science Publishing Co., Inc., New York, NY, 1983), Ehrgott (Multicriteria optimization. Lecture notes in economics and mathematical sciences, vol 491. Springer, Berlin, Heidelberg, 2000), Jahn (Vector optimization: theory, applications, and extensions Springer, Berlin, Heidelberg, 2004), Luc (Theory of vector optimization. Lecture notes in economics and mathematical systems, vol 319. Springer, Berlin, Heidelberg, New York, 1989), and Sawaragi et al. (Theory of multiobjective optimization. Academic, Orlando, 1985).

We briefly recall the vector optimization problem (in short, VOP): Let Y be a topological vector space with a nontrivial closed pointed convex cone C, and X be a vector space. Whenever we use $\mathop{\mathrm{int}}\nolimits (C)$, we assume that $\mathop{\mathrm{int}}\nolimits (C)\neq \emptyset$. We consider the problem

$$\displaystyle\begin{array}{rcl} & \mbox{ minimize }f(x), & {}\\ & \mbox{ subject to }x \in K,& {}\\ \end{array}$$

where

∅ ≠ K ⊆ X is a feasible region
f: X → Y is a objective function
Y is the objective space
x is a decision (variable) vector
X is the decision variable space
$\mathcal{Y}:= f(K)$ is the feasible objective region

In the following sections, we show how VOP (see the above equation) can be converted into several scalar-valued minimization problems, and how solutions of the corresponding scalar-valued minimization problems relate to the efficient solutions of VOP. We present linear and nonlinear scalarization techniques. By means of these methods, we are able to characterize efficient solutions of VOP. The presented procedures are based on the scalarization of VOP, that is, on the principle of transforming VOP into a scalar optimization problem. The scalarization problem in these methods is formulated in a parameterizable way. By varying the parameter, different scalar optimization problems can be generated, and hence, several optimal solutions of such problems can be found. Because a scalarized problem is often easier to solve, there is a huge advantage of using scalarization techniques.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Chapter 5. Vector Variational Inequalities

Abstract

The theory of vector variational inequalities began with the pioneer work of F. Giannessi in 1980 where he extended the classical variational inequality for vector-valued functions in the setting of finite dimensional spaces. He also provided some applications to alternative theorems, quadratic programs and complementarity problems. Since then, a large number of papers have appeared in the literature on different aspects of vector variational inequalities. These references are gathered in the bibliography. Later, it is proved that the theory of vector variational inequalities is a powerful tool to study vector optimization problems. In this chapter, we give an introduction to vector variational inequalities, existence theory of their solutions and some applications to vector optimization problems.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Chapter 6. Linear Scalarization of Vector Variational Inequalities

Abstract

This chapter deals with linear scalarization techniques for vector variational inequality problems and Minty vector variational inequality problems. Such concepts are important for deriving numerical algorithms for solving vector variational inequalities.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Chapter 7. Nonsmooth Vector Variational Inequalities

Abstract

In this chapter, we define different kinds of nonsmooth vector variational inequality problems by means of a bifunction. Several existence results for solutions of these nonsmooth vector variational inequality problems are studied. We give some relations among different kinds of solutions of nonsmooth vector optimization problems and nonsmooth variational inequality problems.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Chapter 8. Generalized Vector Variational Inequalities

Abstract

When the objective function involved in the vector optimization problem is not necessarily differentiable, then the method to solve VOP via corresponding vector variational inequality problems is no longer valid. We need to generalize the vector variational inequality problems for set-valued maps. There are several ways to generalize vector variational inequality problems discussed in chapter “Vector Variational Inequalities”. The main objective of this chapter is to generalize the vector variational inequality problems for set-valued maps and to present the existence results for such generalized vector variational inequality problems with or without monotonicity assumption. We also present some relations between a generalized vector variational inequality problem and a vector optimization problem with a nondifferentiable objective function. Several results of this chapter also hold in the setting of Hausdorff topological vector spaces, but for the sake of convenience, our setting is Banach spaces.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Chapter 9. Vector Equilibrium Problems

Abstract

Motivated by various applications of multi-criteria decision making, extensions of scalar equilibrium problems, discussed in Chap. 1, for the vector case were proposed. Among them, the most investigated problems are vector optimization and vector saddle point. In 1956, Blackwell (Pacific Journal of Mathematics 6(1):1–8, 1956) considered matrix games with vector payoff’s and proved the existence theorem for such problems. Since then, many researchers studied vector non-cooperative games in finite- and infinite-dimensional spaces. In 1980, F. Giannessi extended the variational inequality problem for vector-valued functions known as vector variational inequality problem” (in short, VVIP) with further applications. Vector equilibrium problems (in short, VEPs) can be viewed as further and natural extension of the previous concepts. It is a unified model of several known problems, namely, vector variational inequality problems, vector optimization problems, vector saddle point problems and Nash equilibrium problems for vector-valued functions. The theory of VVIPs and VEPs has been developing extensively since the early ninety’s. In particular, a number of various kinds of these problems were proposed and the corresponding existence results both on bounded and on unbounded sets were established. The mathematical theory of VEPs is presented in this chapter.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Chapter 10. Generalized Vector Equilibrium Problems

Abstract

The main motivation of this chapter is to study the mathematical theory of generalized vector equilibrium problems, namely, existence results for solutions with or without monotonicity, duality and sensitivity analysis.

Qamrul Hasan Ansari, Elisabeth Köbis, Jen-Chih Yao

Backmatter

Titel: Vector Variational Inequalities and Vector Optimization
verfasst von: Prof. Dr. Qamrul Hasan Ansari
Dr. Elisabeth Köbis
Prof. Dr. Jen-Chih Yao
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-63049-6
Print ISBN: 978-3-319-63048-9
DOI: https://doi.org/10.1007/978-3-319-63049-6