Recent Developments in Vector Optimization [electronic resource] /
Ansari, Qamrul Hasan.editor.
Yao, Jen-Chih.editor.
We always come cross several decision-making problems in our daily life. Such problems are always conflicting in which many different view points should be satisfied. In politics, business, industrial systems, management science, networks, etc. one often encounters such kind of problems. The most important and difficult part in such problems is the conflict between various objectives and goals. In these problems, one has to find the minimum(or maximum) for several objective functions. Such problems are called vector optimization problems (VOP),multi-criteria optimization problems or multi-objective optimization problems. This volume deals with several different topics / aspects of vector optimization theory ranging from the very beginning to the most recent one. It contains fourteen chapters written by different experts in the field of vector optimization.
Description based upon print version of record.
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 Problem2.2 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 Case2.5.3 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 Structure4.2.1 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 Scalarizations4.4.2.1 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
Includes bibliographical references and index.
We always come cross several decision-making problems in our daily life. Such problems are always conflicting in which many different view points should be satisfied. In politics, business, industrial systems, management science, networks, etc. one often encounters such kind of problems. The most important and difficult part in such problems is the conflict between various objectives and goals. In these problems, one has to find the minimum(or maximum) for several objective functions. Such problems are called vector optimization problems (VOP),multi-criteria optimization problems or multi-objective optimization problems. This volume deals with several different topics / aspects of vector optimization theory ranging from the very beginning to the most recent one. It contains fourteen chapters written by different experts in the field of vector optimization.
Operations research.
Decision making.
Mathematical optimization.
Management science.
Operations Research/Decision Theory.
Optimization.
Operations Research, Management Science.
Vector Optimization