Recent Developments in Vector Optimization

Name: Recent Developments in Vector Optimization
Brand: Springer
Price: 96.29 EUR
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Qamrul Hasan Ansari Jen-Chih Yao(Herausgeber*in)

Springer (Verlag)

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Erschienen am 21. September 2011

XXIV, 552 Seiten

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Inhalt

Intro
Recent Developments in Vector Optimization
Preface
Contents
Contributors
Acronyms
List of Abbreviations
List of Notations and Symbols
Chapter 1: Vector Optimization Problems and Their Solution Concepts
1.1 Introduction
1.2 Pre-Orders and Partial Orders
1.3 Optimality Concepts in Linear Spaces
1.4 Optimality Concepts in Set Optimization
1.4.1 Vector Approach
1.4.2 Set Approach
1.5 Existence Results in Vector Optimization
1.6 Application: Field Design of a Magnetic Resonance System
References
Chapter 2: Gordan-Type Alternative Theorems and Vector Optimization Revisited
2.1 Introduction and Formulation of the Problem
2.2 Basic Definitions and Preliminaries
2.3 Equivalent Formulations of Gordan-Type Alternative Theorems
2.3.1 Via Quasi-Relative Interior
2.3.2 Via Topological Interior
2.4 A Bidimensional Optimal Alternative Theorem and a Characterization of Two-Dimensionality
2.5 Applications to Vector Optimization
2.5.1 Characterizing Weakly Efficient Solutions Through Linear Scalarization of Bicriteria Problems
2.5.1.1 The Pareto Case
2.5.2 Characterizing Properly Efficient Solutions Through Linear Scalarization of Bicriteria Problems
2.5.2.1 The Pareto Case
2.5.3 Characterizing the Fritz-John Type Optimality Conditions
2.6 More About Proper Efficiency
References
Chapter 3: Duality in Vector Optimization with Infimum and Supremum
3.1 Introduction
3.2 A Complete Lattice for Vector Optimization
3.2.1 Upper Closed and Infimal Sets
3.2.2 The Space of Self-Infimal Sets
3.2.3 Scalarization
3.3 Duality Theory
3.3.1 A General Duality Scheme
3.3.2 Lagrange Duality
3.3.3 Conjugate Duality
3.3.4 Connections to Classic Results
References
Chapter 4: Variable Ordering Structures in Vector Optimization
4.1 Introduction
4.2 Variable Ordering Structure
4.2.1 Optimality Notions
4.2.2 Variable Ordering Structures in Applications
4.2.2.1 Variable Ordering Structures in Medical Image Registration
4.2.2.2 Variable Ordering Structures in Intensity Modulated Radiation Therapy
4.2.2.3 Variable Ordering Structures and Equitability
4.2.3 Vector Variational Inequalities and Vector Complementarity Problems with a Variable Ordering Structure
4.2.4 Variable Ordering Structures Defined by Bishop-Phelps Cones
4.3 Basic Properties of Optimal Elements
4.4 Scalarization
4.4.1 Linear Scalarization
4.4.2 Nonlinear Scalarizations
4.4.2.1 Hiriart-Urruty Scalarization
4.4.2.2 Pascoletti-Serafini Scalarization
4.4.2.3 Scalarization for Ordering Maps with Images Bishop-Phelps Cones
4.5 Duality
References
Chapter 5: Strong KKT, Second Order Conditions and Non-solid Cones in Vector Optimization
5.1 Introduction
5.2 Tools from Nonsmooth Analysis
5.3 Geometric Optimality Conditions
5.4 Optimality Conditions with Explicit Constraints
5.5 Second Order Optimality Conditions
5.6 Excursions in Infinite Dimensions
References
Chapter 6: Optimality Conditions and Image Space Analysis for Vector Optimization Problems
6.1 Introduction
6.2 Generalized Convex Functions and Scalarization Methods
6.2.1 Generalized Convex Functions
6.2.2 Scalarization
6.3 Connections with Vector Variational Inequalities
6.3.1 Optimality Conditions for Differentiable VOP
6.3.2 Optimality Conditions for Nondifferentiable VOP
6.4 Image Space Analysis and Saddle Point Optimality Conditions
6.4.1 Image of VOP
6.4.2 Vector Separation in the Image Space
6.4.3 Scalar Separation in the Image Space
6.4.4 Connections with Duality
6.5 Necessary Optimality Conditions
6.5.1 Necessary Optimality Conditions for Differentiable VOP
6.5.2 Semidifferentiable Problems
References
Chapter 7: Nonsmooth Invexities, Invariant Monotonicities and Nonsmooth Vector Variational-Like Inequalities with Applications to Vector Optimization
7.1 Introduction
7.2 Preliminaries
7.2.1 Convexity
7.2.2 Invexity
7.3 Directional Derivatives
7.4 Nonsmooth Invexities
7.5 Invariant Monotonicities
7.6 Nonsmooth Vector Variational-Like Inequalities
7.7 Nonsmooth Vector Optimization
References
Chapter 8: Optimality Conditions for Approximate Solutions of Convex Semi-Infinite Vector Optimization Problems
8.1 Introduction
8.2 Preliminaries
8.3 Approximate Solutions and Constraint Qualification Conditions
8.4 Optimality Conditions for -Efficient/Efficient Solutions of (CSIVP)
8.5 Optimality Conditions for Weakly -Efficient Solutions of (CSIVP)
References
Chapter 9: Linear Fractional and Convex Quadratic Vector Optimization Problems
9.1 Introduction
9.2 Monotone Affine Variational Inequalities
9.3 Linear Fractional Vector Optimization Problems
9.4 Convex Quadratic Vector Optimization Problems
9.5 Open Problems
9.5.1 Monotone AVVIs
9.5.2 LFVOPs
9.5.3 Convex QVOPs
9.5.4 Strictly Quasiconvex VOPs
References
Chapter 10: Levitin-Polyak Type Well-Posednessin Constrained Optimization
10.1 Introduction
10.2 Generalized Levitin-Polyak Well-Posedness in Constrained Optimization
10.2.1 Preliminaries
10.2.2 Necessary and Sufficient Conditions for (Generalized) LP Well-Posedness
10.2.3 Relations Among Three Types of (Generalized) LP Well-Posedness
10.2.4 Applications to Penalty-Type Methods
10.2.4.1 Penalty Methods
10.2.4.2 Augmented Lagrangian Methods
10.3 Levitin-Polyak Well-Posedness of Constrained Vector Optimization Problems
10.3.1 Preliminaries
10.3.2 Criteria and Characterizations for (Generalized) LP Well-Posedness
10.3.3 Relations Among Various Types of (Generalized) LP Well-Posedness
10.3.4 Application to a Class of Penalty Methods
References
Chapter 11: Vector Variational Principles for Set-Valued Functions
11.1 Introduction
11.2 Preliminaries
11.3 Nonlinear Scalarization Functions
11.3.1 Construction of Scalarizing Functionals
11.3.2 Properties of Scalarization Functions
11.3.3 Continuity Properties
11.3.4 Lipschitz Properties
11.3.5 The Formula for the Conjugate and Subdifferential of ?A for A Convex
11.4 Minimal-Point Theorems and Corresponding Variational Principles
11.4.1 Introduction
11.4.2 Minimal Points in Product Spaces
11.4.3 Minimal-Point Theorems of Isac-Tammer's Type
11.4.4 Ekeland's Variational Principles of Ha's Type
11.4.5 Ekeland's Variational Principle for Bi-Multifunctions
11.4.6 EVP Type Results
11.5 Applications in Vector Optimization
11.5.1 Solution Concepts
11.5.2 Necessary Optimality Conditions in Vector Optimization
References
Chapter 12: The Fermat Rule and Lagrange Multiplier Rule for Various Efficient Solutions of Set-Valued Optimization Problems Expressed in Terms of Coderivatives
12.1 Introduction
12.2 Some Tools from Variational Analysis
12.2.1 Normal Cone, Subdifferential and Coderivative in the Senses of Fréchet, Ioffe, Clarke and Mordukhovich
12.2.2 Sequential Normal Compactness, Pseudo-Lipschitzity and Metric Regularity
12.2.3 Extremal Principle
12.3 Efficient Points of a Set and Efficient Solutions of Set-Valued Optimization Problems
12.3.1 Definitions of Efficient Points of a Set
12.3.2 Concepts of Efficient Solutions of Set-Valued Optimization Problems
12.3.3 A Unified Scalarization Approach to Several Kinds of Efficient Points
12.4 The Fermat Rule
12.4.1 The Fermat Rule for Pareto Efficient Solutions
12.4.2 The Fermat Rule for Strongly, Weakly and Properly Efficient Solutions of (P)
12.5 The Lagrange Multiplier Rule
12.5.1 The Lagrange Multiplier Rule for Pareto Efficient Solutions of (CP)
12.5.2 The Lagrange Multiplier Rule for Weakly, Strongly, Properly Efficient Solutions of (CP)
References
Chapter 13: Extended Pareto Optimality in Multiobjective Problems
13.1 Introduction
13.2 Basic Tools of Variational Analysis
13.3 Necessary Optimality Conditions for Extended Pareto Minimal Points of Sets
13.4 Applications to Set-Valued Optimization
13.5 Multiobjective Optimization with Operator Constraints
References
Chapter 14: Vector Optimization and Cooperative Games
14.1 Introduction
14.2 Fundamentals of Cooperative Games
14.2.1 Cooperative Games
14.2.2 Solutions of Cooperative Games
14.3 Core Solutions in Vector-Valued Games
14.3.1 Vector-Valued Cooperative Games
14.3.2 Solution Concepts with Weak Ordering
14.3.3 Solution Concepts with Strong Ordering
14.4 Partially Ordered Cooperative Games
14.4.1 Partially Ordered Cooperative Games
14.4.2 Core Solutions
14.4.3 The Extended Shapley Value
14.5 Core Solutions in Set-Valued Games
14.5.1 Set-Valued Games and Core Concepts
14.5.2 Existence Theorems
14.6 Multiobjective Games with Restrictions on Coalitions
14.6.1 Maximum and Minimum of a Set in Rm
14.6.2 Multiobjective Cooperative Games
14.6.3 Restricted Multiobjective Cooperative Games by Partition Systems
14.6.4 Inheritance of Convexity
14.6.5 The Core of Restricted Games
14.7 Multiobjective Linear Production Games
14.7.1 Linear Production Games
14.7.2 Multiobjective Linear Production Games
14.7.3 Solutions of the Multiobjective Linear Production Games
14.8 Multiobjective Minimum Cost Spanning Tree Games
14.8.1 Minimum Cost Spanning Tree Games
14.8.2 Multiobjective Minimum Cost Spanning Tree Games
14.8.3 Core Concepts in Multiobjective Minimum Cost Spanning Tree Games
14.9 Conclusion
References
Index

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