Variational Methods in Partially Ordered Spaces (CMS/CAIMS Books in Mathematics, 7)

✍ Scribed by Alfred Göpfert, Hassan Riahi, Christiane Tammer, Constantin Zǎlinescu

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 576
• Category: Library

✦ Synopsis

In mathematical modeling of processes occurring in logistics, management science, operations research, networks, mathematical finance, medicine, and control theory, one often encounters optimization problems involving more than one objective function so that Multiobjective Optimization (or Vector Optimization, initiated by W. Pareto) has received new impetus. The growing interest in vector optimization problems, both from the theoretical point of view and as it concerns applications to real world optimization problems, asks for a general scheme which embraces several existing developments and stimulates new ones.

This book aims to provide the newest results and applications of this quickly growing field. Basic tools of partially ordered spaces are discussed and applied to variational methods in nonlinear analysis and to optimization problems.

The book begins by providing simple examples that illustrate what kind of problems can be handled with the methods presented. The book then deals with connections between order structures and topological structures of sets, discusses properties of nonlinear scalarization functions, and derives corresponding separation theorems for not necessarily convex sets. Furthermore, characterizations of set relations via scalarization are presented.

Important topological properties of multifunctions and new results concerning the theory of vector optimization and equilibrium problems are presented in the book. These results are applied to construct numerical algorithms, especially, proximal-point algorithms and geometric algorithms based on duality assertions.

In the second edition, new sections about set less relations, optimality conditions in set optimization and the asymptotic behavior of multiobjective Pareto-equilibrium problems have been incorporated. Furthermore, a new chapter regarding scalar optimization problems under uncertainty and robust counterpart problems employing approaches based on vector optimization, set optimization, and nonlinear scalarization was added.

Throughout the entire book, there are examples used to illustrate the results and check the stated conditions.

This book will be of interest to graduate students and researchers in pure and applied mathematics, economics, and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book.

✦ Table of Contents

Preface to the First Edition
Mathematical Background
Purpose of This Book
Organization
Preface to the Second Edition
Acknowledgments
Contents
List of Symbols and Abbreviations
Abbreviations
Spaces
Sets
Cones, Relations, Measures, Nets
Sets of Optimal Elements
Functions and Operators
Set-Valued Mappings
List of Figures
1 Examples
1.1 Cones in Vector Spaces
1.2 Equilibrium Problems
1.3 Location Problems in Town Planning
1.4 Multiobjective Control Problems
1.5 Stochastic Dominance
1.6 Uncertainty
2 Functional Analysis over Cones
2.1 Order Structures
2.1.1 Binary relations
2.1.2 Cone order structures on linear spaces
2.2 Functional Analysis and Convexity
2.2.1 Locally convex spaces
2.2.2 Examples and properties of locally convex spaces
2.2.3 Asplund spaces
2.2.4 Special results in the theory of locally convex spaces
2.3 Generalized Set Less Relations
2.4 Separation Theorems for Not Necessarily Convex Sets
2.4.1 Algebraic and Topological Properties
2.4.2 Continuity and Lipschitz Continuity
2.4.3 Separation Properties
2.5 Characterization of set relations by means of nonlinear functionals
2.6 Convexity Notions for Sets and Multifunctions
2.7 Continuity Notions for Multifunctions
2.8 Continuity Notions for Extended Vector-valued Functions
2.9 Extended Multifunctions
2.10 Continuity Properties of Multifunctions Under Convexity Assumptions
2.11 Tangent Cones and Differentiability of Multifunctions
2.12 Radial Epi-Differentiability of Extended Vector-Valued Functions
3 Optimization in Partially Ordered Spaces
3.1 Solution Concepts in Vector Optimization
3.1.1 Approximate Minimality
3.1.2 A General Scalarization Method
3.2 Solution Concepts in Set-Valued Optimization
3.2.1 Solution Concepts Based on Vector Approach
3.2.2 Solution Concepts Based on Set Approach
3.3 Existence Results for Efficient Points
3.3.1 Preliminary Notions and Results Concerning Transitive Relations
3.3.2 Existence of Maximal Elements with Respect to Transitive Relations
3.3.3 Existence of Efficient Points with Respect to Cones
3.3.4 Types of Convex Cones and Compactness with Respect to Cones
3.3.5 Classification of Existence Results for Efficient Points
3.3.6 Some Density and Connectedness Results
3.4 Continuity Properties with Respect to a Scalarization Parameter
3.5 Well-Posedness of Vector Optimization Problems
3.6 Continuity Properties
3.6.1 Continuity Properties of Optimal-Value Multifunctions
3.6.2 Continuity Properties for the Optimal Multifunction in the Case of Moving Cones
3.6.3 Continuity Properties for the Solution Multifunction
3.7 Sensitivity of Vector Optimization Problems
3.8 Duality
3.8.1 Duality Without Scalarization
3.8.2 Duality by Scalarization
3.8.3 Duality for Approximation Problems
3.9 Vector Equilibrium Problems and Related Topics
3.9.1 Vector Equilibrium Problems
3.9.2 General Vector Monotonicity
3.9.3 Generalized KKM Lemma
3.9.4 Existence of Vector Equilibria by Use of the Generalized KKM Lemma
3.9.5 Existence by Scalarization of Vector Equilibrium Problems
3.9.6 Some Knowledge About the Assumptions
3.9.7 Some Particular Cases
3.9.8 Mixed Vector Equilibrium Problems
3.10 Vector Variational Inequalities
3.10.1 Vector Variational-Like Inequalities
3.10.2 Perturbed Vector Variational Inequalities
3.10.3 Hemivariational Inequality Systems
3.10.4 Vector Complementarity Problems
3.10.5 Vector Optimization Problems
3.10.6 Minimax Theorem for Vector-Valued Mappings
3.11 Minimal-Point Theorems in Product Spaces and Corresponding Variational Principles
3.11.1 Not Authentic Minimal-Point Theorems
3.11.2 Authentic Minimal-Point Theorems
3.11.3 Minimal-Point Theorems and Gauge Techniques
3.11.4 Minimal-Point Theorems and Cone-Valued Metrics
3.11.5 Fixed Point Theorems of Kirk–Caristi Type
3.12 Saddle Point Theory
3.12.1 Lagrange Multipliers and Saddle Point Assertions
3.12.2 ε-Saddle Point Assertions
4 Generalized Differentiation and Optimality Conditions
4.1 Mordukhovich/limiting generalized differentiation
4.2 General Concept of Set Extremality
4.2.1 Introduction to Set Extremality
4.2.2 Characterizations of Asplund spaces
4.2.3 Properties and Applications of Extremal systems
4.2.4 Extremality and Optimality
4.3 Subdifferentials of Scalarization Functionals
4.4 Application to Optimization Problems with Set-valued Objectives
5 Applications
5.1 Approximation Problems
5.1.1 General Approximation Problems
5.1.2 Finite-dimensional Approximation Problems
5.1.3 Lp-Approximation Problems
5.1.4 Example: The Inverse Stefan Problem
5.2 Solution Procedures
5.2.1 A Proximal-Point Algorithm for Real-Valued Control Approximation Problems
5.2.2 An Interactive Algorithm for the Vector Control Approximation Problem
5.2.3 Proximal Algorithms for Vector Equilibrium Problems
5.2.4 Relaxation and Penalization for Vector Equilibrium Problems
5.3 Location Problems
5.3.1 Formulation of the Problem
5.3.2 An Algorithm for the Multiobjective Location Problem
5.4 Multiobjective Control Problems
5.4.1 The Formulation of the Problem
5.4.2 An ε-Minimum Principle for Multiobjective Optimal Control Problems
5.4.3 A Multiobjective Stochastic Control Problem
5.5 To attain Pareto-equilibria by Asymptotic Behavior …
5.5.1 Pareto Equilibria and Associate Critical Points
5.5.2 Existence of Solutions for (WVEP) and (LSEP)λ
5.5.3 Existence of Solutions for Continuous First-Order Equilibrium Dynamical Systems
5.5.4 Asymptotic Behavior of Solutions when t rightarrow+infty
5.5.5 Asymptotic Behavior for Multiobjective Optimization Problem
5.5.6 Asymptotic Behavior for Multiobjective Saddle-point Problem
5.5.7 Numerical examples
6 Scalar Optimization under Uncertainty
6.1 Robustness and Stochastic Programming
6.2 Scalar Optimization under Uncertainty
6.2.1 Formulation of Optimization Problems under Uncertainty
6.2.2 Strict Robustness
6.2.3 Optimistic Robustness
6.2.4 Regret Robustness
6.2.5 Reliability
6.2.6 Adjustable Robustness
6.2.7 Minimizing the Expectation
6.2.8 Stochastic Dominance
6.2.9 Two-Stage Stochastic Programming
6.3 Vector Optimization as Unifying Concept
6.3.1 Vector Approach for Strict Robustness
6.3.2 Vector Approach for Optimistic Robustness
6.3.3 Vector Approach for Regret Robustness
6.3.4 Vector Approach for Reliability
6.3.5 Vector Approach for Adjustable Robustness
6.3.6 Vector Approach for Minimizing the Expectation
6.3.7 Vector Approach for Stochastic Dominance
6.3.8 Vector Approach for Two-stage Stochastic Programming
6.3.9 Proper Robustness
6.3.10 An Overview on Concepts of Robustness Based on Vector Approach
6.4 Set Relations as Unifying Concept
6.4.1 Set Approach for Strict Robustness
6.4.2 Set Approach for Optimistic Robustness
6.4.3 Set Approach for Regret Robustness
6.4.4 Set Approach for Reliability
6.4.5 Set Approach for Adjustable Robustness
6.4.6 Certain Robustness
6.4.7 An Overview on Concepts of Robustness Based on Set Relations
6.5 Translation Invariant Functions as Unifying Concept
6.5.1 Nonlinear Scalarizing Functional for Strict Robustness
6.5.2 Nonlinear Scalarizing Functional for Optimistic Robustness
6.5.3 Nonlinear Scalarizing Functional for Regret Robustness
6.5.4 Nonlinear Scalarizing Functional for Reliability
6.5.5 Nonlinear Scalarizing Functional for Adjustable Robustness
6.5.6 Nonlinear Scalarizing Functional for Minimizing the Expectation
6.5.7 Nonlinear Scalarizing Functional for Two-Stage Stochastic Programming
6.5.8 Nonlinear Scalarization Approach for ε-Constraint Robustness
6.5.9 An Overview on Concepts of Robustness Based on Translation Invariant Functionals
References
Index