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 given. New numerical methods for efficiently solving set optimization problems are provided. Moreover, applications in economics, finance and risk theory are discussed.
The objective of this book is to present advances in different areas of variational analysis and set optimization, especially uncertain optimization, optimal control and bilevel optimization. Uncertain optimization problems will be approached from both a stochastic as well as a robust point of view. This leads to different interpretations of the solutions, which widens the choices for a decision-maker given his preferences.
Recent developments regarding linear and nonlinear scalarization techniques with solid and nonsolid ordering cones for solving set optimization problems are discussed in this book. These results are useful for deriving optimality conditions for set and vector optimization problems.
Consequently, necessary and sufficient optimality conditions are presented within this book, both in terms of scalarization as well as generalized derivatives. Moreover, an overview of existing duality statements and new duality assertions is given.
The book also addresses the field of variable domination structures in vector and set optimization. Including variable ordering cones is especially important in applications such as medical image registration with uncertainties.
This book covers a wide range of applications of set optimization. These range from finance, investment, insurance, control theory, economics to risk theory. As uncertain multi-objective optimization, especially robust approaches, lead to set optimization, one main focus of this book is uncertain optimization.
Important recent developments concerning numerical methods for solving set optimization problems sufficiently fast are main features of this book. These are illustrated by various examples as well as easy-to-follow-steps in order to facilitate the decision process for users. Simple techniques aimed at practitioners working in the fields of mathematical programming, finance and portfolio selection are presented. These will help in the decision-making process, as well as give an overview of nondominated solutions to choose from.
Table of Contents
Variational Analysis and Variational Rationality in Behavioral Sciences
Boris S. Mordukhovich and Antoine Soubeyran
Variational Rationality in Behavioral Sciences
Evaluation Aspects of Variational Rationality
Exact Stationary Traps in Behavioral Dynamics
Evaluations of Approximate Stationary Traps
Geometric Evaluations and Extremal Principle
Summary of Major Finding and Future Research
A Financial Model for a Multi-Period Portfolio Optimization Problem
Gabriella Colajanni and Patrizia Daniele
The Financial Model
Variational Inequality Formulation and Existence Results
A Generalized Proximal Alternating Linearized Method
Antoine Soubeyran, Jo˜ao Carlos Souza, and Jo˜ao Xavier Cruz Neto
Potential Games: How to Play Nash?
Variational Analysis: How to Optimize a Potential Function?
Variational Rationality: How Human Dynamics Work?
Computing How to Play Nash for Potential Games
Sublinear-like Scalarization Scheme for Sets and its Applications
Koichiro Ike, Yuto Ogata, Tamaki Tanaka, and Hui Yu
Set Relations and Scalarizing Functions for Sets
Inherited Properties of Scalarizing Functions
Applications to Set-valued Inequality and Fuzzy Theory
Functions with Uniform Sublevel Sets, Epigraphs and Continuity
Directional Closedness of Sets
Definition of Functions with Uniform Sublevel Sets
Nontranslative Functions with Uniform Sublevel Sets
Extension of Arbitrary Functionals to Translative Functions
Optimality and Viability Conditions for State-Constrained Control Problems
Strict Normality and the Decrease Condition
Metric Regularity, Viability, and the Maximum Principle
Lipschitz Properties of Cone-convex Set-valued Functions
Vu Anh Tuan and Thanh Tam Le
Concepts on Convexity and Lipschitzianity of Set-valued Functions
Lipschitz Properties of Cone-convex Set-valued Functions
Vector Optimization with Variable Ordering Structures
Marius Durea, Elena-Andreea Florea, and Radu Strugariu
Sufficient Conditions for Mixed Openness
Necessary Optimality Conditions
Bibliographic Notes, Comments, and Conclusions
Vectorial Penalization in Multi-objective Optimization
Preliminaries in Generalized Convex Multi-objective Optimization
Pareto Efficiency with Respect to Different Constraint Sets
A Vectorial Penalization Approach in Multi-objective Optimization
Penalization in Multi-objective Optimization with Functional
Set Optimization Problems Reducible to Vector Optimization Problems
Gabriele Eichfelder and Tobias Gerlach
Basics of Vector and Set Optimization
Set Optimization Problems Being Reducible to Vector Optimization Problems
Implication on Set-valued Test Instances
Abstract Convexity and Solvability Theorems
Ali Reza Doagooei
Abstract Convex Functions
Solvability Theorems for Real-valued Systems of Inequalities
Vector-valued Abstract Convex Functions and Solvability Theorems
Applications in Optimization
Regularization Methods for Scalar and Vector Control Problems
Baasansuren Jadamba, Akhtar A. Khan, Miguel Sama, and Christiane Tammer
Integral Constraint Regularization
A Constructible Dilating Regularization
Regularization of Vector Optimization Problems
Concluding Remarks and Future Research
Akhtar Khan is a Professor at Rochester Institute of Technology. His has published more than seventy papers on set-valued optimization, inverse problems, and variational inequalities. He is a co-author of Set-valued Optimization, Springer (2015), and Co-editor of Nonlinear Analysis and Variational Problems, Springer (2009). He is Co-Editor in Chief of the Journal of Applied and Numerical Optimization, and Editorial Board member of Optimization, Journal of Optimization Theory and Applications, and Journal of Nonlinear and Variational Analysis.
Elisabeth Köbis is a lecturer and researcher at Martin-Luther-University Halle-Wittenberg, Germany. She received her PhD from Martin-Luther-University Halle-Wittenberg, Germany, in 2014. Her research interests lie in vector and set optimization and its applications to uncertain optimization, in particular robust approaches to uncertain multi-objective optimization problems, and unified approaches to uncertain optimization using nonlinear scalarization, vector variational inequalities and variable domination structures.
Christiane Tammer is working on the field variational analysis and optimization. She has co-authored 4 monographs, i.e. Set-valued Optimization - An Introduction with Applications. Springer (2015), Variational Methods in Partially Ordered Spaces. Springer (2003), Angewandte Funktionalanalysis. Vieweg+Teubner (2009), Approximation und Nichtlineare Optimierung in Praxisaufgaben. Springer (2017). She is the Editor in Chief of the journal Optimization and a member of the Editorial Board of several journals, the Scientific Committee of the Working Group on Generalized Convexity and EUROPT Managing Board.