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 given. New numerical methods for efficiently solving set optimization problems are provided. Moreover, applications in economics, finance and risk theory are discussed. Summary 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.

Preface Variational Analysis and Variational Rationality in Behavioral Sciences Boris S. Mordukhovich and Antoine Soubeyran Introduction 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 References A Financial Model for a Multi-Period Portfolio Optimization Problem Gabriella Colajanni and Patrizia Daniele Introduction The Financial Model Variational Inequality Formulation and Existence Results Numerical Examples Conclusions References A Generalized Proximal Alternating Linearized Method Antoine Soubeyran, Jo˜ao Carlos Souza, and Jo˜ao Xavier Cruz Neto Introduction 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 References Sublinear-like Scalarization Scheme for Sets and its Applications Koichiro Ike, Yuto Ogata, Tamaki Tanaka, and Hui Yu Introduction Set Relations and Scalarizing Functions for Sets Inherited Properties of Scalarizing Functions Applications to Set-valued Inequality and Fuzzy Theory References Functions with Uniform Sublevel Sets, Epigraphs and Continuity Petra Weidner Introduction Preliminaries Directional Closedness of Sets Definition of Functions with Uniform Sublevel Sets Translative Functions Nontranslative Functions with Uniform Sublevel Sets Extension of Arbitrary Functionals to Translative Functions References Optimality and Viability Conditions for State-Constrained Control Problems Robert Kipka Introduction Background Strict Normality and the Decrease Condition Metric Regularity, Viability, and the Maximum Principle Closing Remarks References Lipschitz Properties of Cone-convex Set-valued Functions Vu Anh Tuan and Thanh Tam Le Introduction Preliminaries Concepts on Convexity and Lipschitzianity of Set-valued Functions Lipschitz Properties of Cone-convex Set-valued Functions Conclusions References Vector Optimization with Variable Ordering Structures Marius Durea, Elena-Andreea Florea, and Radu Strugariu Introduction Preliminaries Efficiency Concepts Sufficient Conditions for Mixed Openness Necessary Optimality Conditions Bibliographic Notes, Comments, and Conclusions References Vectorial Penalization in Multi-objective Optimization Christian G¨unther Introduction 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 Conclusions References Set Optimization Problems Reducible to Vector Optimization Problems Gabriele Eichfelder and Tobias Gerlach Introduction Basics of Vector and Set Optimization Set Optimization Problems Being Reducible to Vector Optimization Problems Implication on Set-valued Test Instances References Abstract Convexity and Solvability Theorems Ali Reza Doagooei Introduction Abstract Convex Functions Solvability Theorems for Real-valued Systems of Inequalities Vector-valued Abstract Convex Functions and Solvability Theorems Applications in Optimization References Regularization Methods for Scalar and Vector Control Problems Baasansuren Jadamba, Akhtar A. Khan, Miguel Sama, and Christiane Tammer Introduction Lavrentiev Regularization Conical Regularization Half-space Regularization Integral Constraint Regularization A Constructible Dilating Regularization Regularization of Vector Optimization Problems Concluding Remarks and Future Research References

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.