Variational Analysis and Set Optimization : Developments and Applications in Decision Making book cover
1st Edition

Variational Analysis and Set Optimization
Developments and Applications in Decision Making

ISBN 9781138037267
Published June 7, 2019 by CRC Press
348 Pages 9 B/W Illustrations

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Book Description

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

Numerical Examples




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

Petra Weidner



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



Optimality and Viability Conditions for State-Constrained Control Problems

Robert Kipka



Strict Normality and the Decrease Condition

Metric Regularity, Viability, and the Maximum Principle

Closing Remarks



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



Efficiency Concepts

Sufficient Conditions for Mixed Openness

Necessary Optimality Conditions

Bibliographic Notes, Comments, and Conclusions











Vectorial Penalization in Multi-objective Optimization

Christian G¨unther


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


Lavrentiev Regularization

Conical Regularization

Half-space Regularization

Integral Constraint Regularization

A Constructible Dilating Regularization

Regularization of Vector Optimization Problems

Concluding Remarks and Future Research


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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.