Variational analysis and set optimization developments and applications in decision making
This book contains the latest advances in variational analysis and set / vector optimization, including uncertain optimization, optimal control and bilevel optimization. Recent developments concerning scalarization techniques, necessary and sufficient optimality conditions and duality statements are...
| Otros Autores: | , , |
|---|---|
| Formato: | Libro electrónico |
| Idioma: | Inglés |
| Publicado: |
Boca Raton, FL :
CRC Press
2019.
|
| Edición: | 1st ed |
| Materias: | |
| Ver en Biblioteca Universitat Ramon Llull: | https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009669536506719 |
Tabla de Contenidos:
- Cover
- Title Page
- Copyright Page
- Dedication
- Preface
- Table of Contents
- 1: Variational Analysis and Variational Rationality in Behavioral Sciences: Stationary Traps
- 1.1 Introduction
- 1.2 Variational Rationality in Behavioral Sciences
- 1.2.1 Desirability and Feasibility Aspects of Human Behavior
- 1.2.2 Worthwhile (Desirable and Feasible) Moves
- 1.3 Evaluation Aspects of Variational Rationality
- 1.3.1 Optimistic and Pessimistic Evaluations
- 1.3.2 Optimistic Evaluations of Reference-Dependent Payoffs
- 1.4 Exact Stationary Traps in Behavioral Dynamics
- 1.5 Evaluations of Approximate Stationary Traps
- 1.6 Geometric Evaluations and Extremal Principle
- 1.7 Summary of Major Finding and Future Research
- References
- 2: A Financial Model for a Multi-Period Portfolio Optimization Problem with a Variational Formulation
- 2.1 Introduction
- 2.2 The Financial Model
- 2.3 Variational Inequality Formulation and Existence Results
- 2.4 Numerical Examples
- 2.5 Conclusions
- References
- 3: How Variational Rational Agents Would Play Nash: A Generalized Proximal Alternating Linearized Method
- 3.1 Introduction
- 3.2 Potential Games: How to Play Nash?
- 3.2.1 Static Potential Games
- 3.2.1.1 Non Cooperative Normal Form Games
- 3.2.1.2 Examples
- 3.2.1.3 Potential Games
- 3.2.2 Dynamic Potential Games
- 3.2.2.1 Alternating Moves and Delays
- 3.2.2.2 The ''Learning How to Play Nash" Problem
- 3.3 Variational Analysis: How to Optimize a Potential Function?
- 3.3.1 Minimizing a Function of Several Variables: Gauss-Seidel Algorithm
- 3.3.2 The Problem of Minimizing a Sum of Two Functions Without a Coupling Term
- 3.3.3 Minimizing a Sum of Functions With a Coupling Term (Potential Function)
- 3.3.3.1 Mathematical Perspective Proximal Regularization of a Gauss-Seidel Algorithm.
- 3.3.3.2 Game Perspective: How to Play Nash in Alternation
- 3.3.3.3 Cross Fertilization Between Game and Mathematical Perspectives
- 3.4 Variational Rationality: How Human Dynamics Work?
- 3.4.1 Stay/stability and Change Dynamics
- 3.4.2 Worthwhile Changes
- 3.4.2.1 One Agent
- 3.4.2.2 Two Interrelated Agents
- 3.4.3 Worthwhile Transitions
- 3.4.4 Ends as Variational Traps
- 3.4.4.1 One Agent
- 3.4.4.2 Two Interrelated Agents
- 3.5 Computing How to Play Nash for Potential Games
- 3.5.1 Linearization of a Potential Game with Costs to Move as Quasi Distances
- References
- 4: Sublinear-like Scalarization Scheme for Sets and its Applications to Set-valued Inequalities
- 4.1 Introduction
- 4.2 Set Relations and Scalarizing Functions for Sets
- 4.3 Inherited Properties of Scalarizing Functions
- 4.4 Applications to Set-valued Inequality and Fuzzy Theory
- 4.4.1 Set-valued Fan-Takahashi Minimax Inequality
- 4.4.2 Set-valued Gordan-type Alternative Theorems
- 4.4.3 Application to Fuzzy Theory
- References
- 5: Functions with Uniform Sublevel Sets, Epigraphs and Continuity
- 5.1 Introduction
- 5.2 Preliminaries
- 5.3 Directional Closedness of Sets
- 5.4 Definition of Functions with Uniform Sublevel Sets
- 5.5 Translative Functions
- 5.6 Nontranslative Functions with Uniform Sublevel Sets
- 5.7 Extension of Arbitrary Functionals to Translative Functions
- References
- 6: Optimality and Viability Conditions for State-Constrained Optimal Control Problems
- 6.1 Introduction
- 6.1.1 Statement of Problem and Contributions
- 6.1.2 Standing Hypotheses
- 6.2 Background
- 6.2.1 Elements of Nonsmooth Analysis
- 6.2.2 Relaxed Controls
- 6.3 Strict Normality and the Decrease Condition
- 6.3.1 Overview of the Approach Taken
- 6.3.2 The Decrease Condition
- 6.4 Metric Regularity, Viability, and the Maximum Principle.
- 6.5 Closing Remarks
- References
- 7: Lipschitz Properties of Cone-convex Set-valued Functions
- 7.1 Introduction
- 7.2 Preliminaries
- 7.3 Concepts on Convexity and Lipschitzianity of Set-valued Functions
- 7.3.1 Set Relations and Set Differences
- 7.3.2 Cone-convex Set-valued Functions
- 7.3.3 Lipschitz Properties of Set-valued Functions
- 7.4 Lipschitz Properties of Cone-convex Set-valued Functions
- 7.4.1 (C, e)-Lipschitzianity
- 7.4.2 C-Lipschitzianity
- 7.4.3 G-Lipschitzianity
- 7.5 Conclusions
- References
- 8: Efficiencies and Optimality Conditions in Vector Optimization with Variable Ordering Structures
- 8.1 Introduction
- 8.2 Preliminaries
- 8.3 Efficiency Concepts
- 8.4 Sufficient Conditions for Mixed Openness
- 8.5 Necessary Optimality Conditions
- 8.6 Bibliographic Notes, Comments, and Conclusions
- References
- 9: Vectorial Penalization in Multi-objective Optimization
- 9.1 Introduction
- 9.2 Preliminaries in Generalized Convex Multi-objective Optimization
- 9.3 Pareto Efficiency With Respect to Different Constraint Sets
- 9.4 A Vectorial Penalization Approach in Multi-objective Optimization
- 9.4.1 Method by Vectorial Penalization
- 9.4.2 Main Relationships
- 9.5 Penalization in Multi-objective Optimization with Functional Inequality Constraints
- 9.5.1 The Case of a Not Necessarily Convex Feasible Set
- 9.5.2 The Case of a Convex Feasible Set But Without Convex Representation
- 9.6 Conclusions
- References
- 10: On Classes of Set Optimization Problems which are Reducible to Vector Optimization Problems and its Impact on Numerical Test Instances
- 10.1 Introduction
- 10.2 Basics of Vector and Set Optimization
- 10.3 Set Optimization Problems Being Reducible to Vector Optimization Problems
- 10.3.1 Set-valued Maps Based on a Fixed Set
- 10.3.2 Box-valued Maps
- 10.3.3 Ball-valued Maps.
- 10.4 Implication on Set-valued Test Instances
- References
- 11: Abstract Convexity and Solvability Theorems
- 11.1 Introduction
- 11.2 Abstract Convex Functions
- 11.3 Solvability Theorems for Real-valued Systems of Inequalities
- 11.3.1 Polar Functions of IPH and ICR Functions
- 11.3.2 Solvability Theorems for IPH and ICR Functions
- 11.3.3 Solvability Theorem for Topical Functions
- 11.4 Vector-valued Abstract Convex Functions and Solvability Theorems
- 11.4.1 Vector-valued IPH Functions and Solvability Theorems
- 11.4.2 Vector-valued ICR Functions and Solvability Theorems
- 11.4.3 Vector-valued Topical Functions and Solvability Theorems
- 11.5 Applications in Optimization
- 11.5.1 IPH and ICR Maximization Problems
- 11.5.2 A New Approach to Solve Linear Programming Problems with Nonnegative Multipliers
- References
- 12: Regularization Methods for Scalar and Vector Control Problems
- 12.1 Introduction
- 12.2 Lavrentiev Regularization
- 12.3 Conical Regularization
- 12.4 Half-space Regularization
- 12.5 Integral Constraint Regularization
- 12.6 A Constructible Dilating Regularization
- 12.7 Regularization of Vector Optimization Problems
- 12.8 Concluding Remarks and Future Research
- 12.8.1 Conical Regularization for Variational Inequalities
- 12.8.2 Applications to Supply Chain Networks
- 12.8.3 Nonlinear Scalarization for Vector Optimal Control Problems
- 12.8.4 Nash Equilibrium Leading to Variational Inequalities
- References
- Index.
