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E-raamat: Variational Analysis and Set Optimization: Developments and Applications in Decision Making

Edited by (Martin-Luther-University, Germany), Edited by (Martin-Luther-University Halle-Wittenberg, Germany), Edited by (Rochester Institute of Technology, New York, USA)
  • Formaat: 348 pages
  • Ilmumisaeg: 07-Jun-2019
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351712064
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Variational Analysis and Set Optimization: Developments and Applications in Decision MakingSuurem pilt
  • Formaat: 348 pages
  • Ilmumisaeg: 07-Jun-2019
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351712064
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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 v
1 Variational Analysis and Variational Rationality in Behavioral Sciences: Stationary Traps
1(24)
Boris S. Mordukhovich
Antoine Soubeyran
1.1 Introduction
1(2)
1.2 Variational Rationality in Behavioral Sciences
3(2)
1.2.1 Desirability and Feasibility Aspects of Human Behavior
3(1)
1.2.2 Worthwhile (Desirable and Feasible) Moves
4(1)
1.3 Evaluation Aspects of Variational Rationality
5(6)
1.3.1 Optimistic and Pessimistic Evaluations
5(4)
1.3.2 Optimistic Evaluations of Reference-Dependent Payoffs
9(2)
1.4 Exact Stationary Traps in Behavioral Dynamics
11(2)
1.5 Evaluations of Approximate Stationary Traps
13(4)
1.6 Geometric Evaluations and Extremal Principle
17(3)
1.7 Summary of Major Finding and Future Research
20(5)
References
22(3)
2 A Financial Model for a Multi-Period Portfolio Optimization Problem with a Variational Formulation
25(19)
Gabriella Colajanni
Patrizia Daniele
2.1 Introduction
25(3)
2.2 The Financial Model
28(7)
2.3 Variational Inequality Formulation and Existence Results
35(4)
2.4 Numerical Examples
39(2)
2.5 Conclusions
41(3)
References
42(2)
3 How Variational Rational Agents Would Play Nash: A Generalized Proximal Alternating Linearized Method
44(28)
Antoine Soubeyran
Joao Carlos Souza
Joao Xavier Cruz Neto
3.1 Introduction
44(1)
3.2 Potential Games: How to Play Nash?
45(5)
3.2.1 Static Potential Games
45(1)
3.2.1.1 Non Cooperative Normal Form Games
46(1)
3.2.1.2 Examples
46(2)
3.2.1.3 Potential Games
48(1)
3.2.2 Dynamic Potential Games
49(1)
3.2.2.1 Alternating Moves and Delays
49(1)
3.2.2.2 The "Learning How to Play Nash" Problem
49(1)
3.3 Variational Analysis: How to Optimize a Potential Function?
50(4)
3.3.1 Minimizing a Function of Several Variables: Gauss-Seidel Algorithm
50(1)
3.3.2 The Problem of Minimizing a Sum of Two Functions Without a Coupling Term
51(1)
3.3.3 Minimizing a Sum of Functions With a Coupling Term (Potential Function)
52(1)
3.3.3.1 Mathematical Perspective Proximal Regularization of a Gauss-Seidel Algorithm
52(1)
3.3.3.2 Game Perspective: How to Play Nash in Alternation
52(1)
3.3.3.3 Cross Fertilization Between Game and Mathematical Perspectives
53(1)
3.4 Variational Rationality: How Human Dynamics Work?
54(5)
3.4.1 Stay/stability and Change Dynamics
54(1)
3.4.2 Worthwhile Changes
54(1)
3.4.2.1 One Agent
55(1)
3.4.2.2 Two Interrelated Agents
56(1)
3.4.3 Worthwhile Transitions
57(1)
3.4.4 Ends as Variational Traps
58(1)
3.4.4.1 One Agent
58(1)
3.4.4.2 Two Interrelated Agents
59(1)
3.5 Computing How to Play Nash for Potential Games
59(13)
3.5.1 Linearization of a Potential Game with Costs to Move as Quasi Distances
59(9)
References
68(4)
4 Sublinear-like Scalarization Scheme for Sets and its Applications to Set-valued Inequalities
72(20)
Koichiro Ike
Yuto Ogata
Tamaki Tanaka
Hui Yu
4.1 Introduction
72(2)
4.2 Set Relations and Scalarizing Functions for Sets
74(9)
4.3 Inherited Properties of Scalarizing Functions
83(2)
4.4 Applications to Set-valued Inequality and Fuzzy Theory
85(7)
4.4.1 Set-valued Fan-Takahashi Minimax Inequality
85(2)
4.4.2 Set-valued Gordan-type Alternative Theorems
87(1)
4.4.3 Application to Fuzzy Theory
88(2)
References
90(2)
5 Functions with Uniform Sublevel Sets, Epigraphs and Continuity
92(20)
Petra Weidner
5.1 Introduction
92(1)
5.2 Preliminaries
93(1)
5.3 Directional Closedness of Sets
94(3)
5.4 Definition of Functions with Uniform Sublevel Sets
97(1)
5.5 Translative Functions
98(5)
5.6 Nontranslative Functions with Uniform Sublevel Sets
103(3)
5.7 Extension of Arbitrary Functionals to Translative Functions
106(6)
References
110(2)
6 Optimality and Viability Conditions for State-Constrained Optimal Control Problems
112(17)
Robert Kipka
6.1 Introduction
112(3)
6.1.1 Statement of Problem and Contributions
113(2)
6.1.2 Standing Hypotheses
115(1)
6.2 Background
115(4)
6.2.1 Elements of Nonsmooth Analysis
115(2)
6.2.2 Relaxed Controls
117(2)
6.3 Strict Normality and the Decrease Condition
119(4)
6.3.1 Overview of the Approach Taken
119(1)
6.3.2 The Decrease Condition
120(3)
6.4 Metric Regularity, Viability, and the Maximum Principle
123(2)
6.5 Closing Remarks
125(4)
References
126(3)
7 Lipschitz Properties of Cone-convex Set-valued Functions
129(29)
Vu Anh Tuan
Thanh Tam Le
7.1 Introduction
129(1)
7.2 Preliminaries
130(4)
7.3 Concepts on Convexity and Lipschitzianity of Set-valued Functions
134(11)
7.3.1 Set Relations and Set Differences
134(4)
7.3.2 Cone-convex Set-valued Functions
138(3)
7.3.3 Lipschitz Properties of Set-valued Functions
141(4)
7.4 Lipschitz Properties of Cone-convex Set-valued Functions
145(9)
7.4.1 (C, e)-Lipschitzianity
145(2)
7.4.2 C-Lipschitzianity
147(5)
7.4.3 G-Lipschitzianity
152(2)
7.5 Conclusions
154(4)
References
155(3)
8 Efficiencies and Optimally Conditions in Vector Optimization with Variable Ordering Structures
158(52)
Marius Durea
Elena-Andreea Florea
Radu Strugariu
8.1 Introduction
158(2)
8.2 Preliminaries
160(7)
8.3 Efficiency Concepts
167(13)
8.4 Sufficient Conditions for Mixed Openness
180(12)
8.5 Necessary Optimally Conditions
192(11)
8.6 Bibliographic Notes, Comments, and Conclusions
203(7)
References
206(4)
9 Vectorial Penalization in Multi-objective Optimization
210(31)
Christian Gunther
9.1 Introduction
210(2)
9.2 Preliminaries in Generalized Convex Multi-objective Optimization
212(3)
9.3 Pareto Efficiency With Respect to Different Constraint Sets
215(3)
9.4 A Vectorial Penalization Approach in Multi-objective Optimization
218(10)
9.4.1 Method by Vectorial Penalization
219(3)
9.4.2 Main Relationships
222(6)
9.5 Penalization in Multi-objective Optimization with Functional Inequality Constraints
228(9)
9.5.1 The Case of a Not Necessarily Convex Feasible Set
229(5)
9.5.2 The Case of a Convex Feasible Set But Without Convex Representation
234(3)
9.6 Conclusions
237(4)
References
238(3)
10 On Classes of Set Optimization Problems which are Reducible to Vector Optimization Problems and its Impact on Numerical Test Instances
241(25)
Gabriele Eichfelder
Tobias Gerlach
10.1 Introduction
241(2)
10.2 Basics of Vector and Set Optimization
243(3)
10.3 Set Optimization Problems Being Reducible to Vector Optimization Problems
246(12)
10.3.1 Set-valued Maps Based on a Fixed Set
246(5)
10.3.2 Box-valued Maps
251(3)
10.3.3 Ball-valued Maps
254(4)
10.4 Implication on Set-valued Test Instances
258(8)
References
264(2)
11 Abstract Convexity and Solvability Theorems
266(30)
Ali Reza Doagooei
11.1 Introduction
266(2)
11.2 Abstract Convex Functions
268(4)
11.3 Solvability Theorems for Real-valued Systems of Inequalities
272(6)
11.3.1 Polar Functions of IPH and ICR Functions
273(1)
11.3.2 Solvability Theorems for IPH and ICR Functions
274(3)
11.3.3 Solvability Theorem for Topical Functions
277(1)
11.4 Vector-valued Abstract Convex Functions and Solvability Theorems
278(7)
11.4.1 Vector-valued IPH Functions and Solvability Theorems
279(2)
11.4.2 Vector-valued ICR Functions and Solvability Theorems
281(2)
11.4.3 Vector-valued Topical Functions and Solvability Theorems
283(2)
11.5 Applications in Optimization
285(11)
11.5.1 IPH and ICR Maximization Problems
285(4)
11.5.2 A New Approach to Solve Linear Programming Problems with Nonnegative Multipliers
289(5)
References
294(2)
12 Regularization Methods for Scalar and Vector Control Problems
296(27)
Baasansuren Jadamba
Akhtar A. Khan
Miguel Sama
Christiam Tammer
12.1 Introduction
296(3)
12.2 Lavrentiev Regularization
299(3)
12.3 Conical Regularization
302(2)
12.4 Half-space Regularization
304(2)
12.5 Integral Constraint Regularization
306(3)
12.6 A Constructible Dilating Regularization
309(3)
12.7 Regularization of Vector Optimization Problems
312(2)
12.8 Concluding Remarks and Future Research
314(9)
12.8.1 Conical Regularization for Variational Inequalities
314(1)
12.8.2 Applications to Supply Chain Networks
315(1)
12.8.3 Nonlinear Scalarization for Vector Optimal Control Problems
315(1)
12.8.4 Nash Equilibrium Leading to Variational Inequalities
315(1)
References
316(7)
Index 323
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 andEUROPT Managing Board.
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