Variational Analysis and Set Optimization

Developments and Applications in Decision Making

Akhtar A. Khan, Elisabeth Köbis, Christiane Tammer, Akhtar A. Khan, Elisabeth Köbis, Christiane Tammer

• Publisher: CRC Press
• Year: 2019
• eBook ISBN: 9781351712064
• Edition: 1
• Topic: Betriebswirtschaft
• Subtopic: Operations

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.

Chapter 1

Variational Analysis and Variational Rationality in Behavioral Sciences: Stationary Traps

Boris S. Mordukhovich

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. [email protected]

Antoine Soubeyran

Aix-Marseille School of Economics, Aix-Marseille University, CNRS and EHESS, Marseille 13002, France. [email protected]

1.1 Introduction

Recent years have witnessed broad applications of advanced tools of variational analysis, generalized differentiation, and multiobjective (vector and set-valued) optimization to real-life models, particularly those related to economics and finance; see, e.g., [1, 3, 8, 10, 11, 12] and the references therein. Lately [4, 5], certain variational principles and techniques have been developed and applied to models of behavioral sciences that mainly concern human behavior. The latter applications are based on the variational rationality approach to behavioral sciences initiated by Soubeyran in [17, 18, 19, 20]. Major concepts of the variational rationality approach include the notions of (stationary and variational) traps, which describe underlying positions of individual or group behavior related to making worthwhile decisions on changing or staying at the current position; see Section 2 for more details. Mathematically, these notions correspond to points of equilibria, optima, aspirations, etc., and thus call to employ and develop powerful machinery of variational analysis and optimization theory for their comprehensive study and applications.

Papers [4, 5] mostly dealt with the study of global variational traps in connection with dynamical aspects of the variational rationality approach and its applications to goal systems in psychology [4] and to capability theory of wellbeing in behavioral sciences [5]. Appropriated tools of variational analysis developed in [4, 5] for the study and applications of the aforementioned global dynamical issues were related to set-valued extensions of the Ekeland variational principle in the case of set-valued mappings defined on quasimetric spaces and taken values in vector spaces with variable ordering structures; see also [1, 8, 10] for multiobjective optimization problems of this type.

The main goal of this paper is largely different from those in [4, 5]. Here, we primarily focus on the study and variational analysis descriptions of local stationary traps in the variational rationality framework. It will be shown that the well-understood subdifferential notions for convex and nonconvex extended-real-valued functions, as well as generalized normals to locally closed sets, are instrumental in describing and characterizing various types of such traps. Furthermore, the fundamental variational and extremal principl...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Preface
• Table of Contents
• 1. Variational Analysis and Variational Rationality in Behavioral Sciences: Stationary Traps
• 2. A Financial Model for a Multi-Period Portfolio Optimization Problem with a Variational Formulation
• 3. How Variational Rational Agents Would Play Nash: A Generalized Proximal Alternating Linearized Method
• 4. Sublinear-like Scalarization Scheme for Sets and its Applications to Set-valued Inequalities
• 5. Functions with Uniform Sublevel Sets, Epigraphs and Continuity
• 6. Optimality and Viability Conditions for State-Constrained Optimal Control Problems
• 7. Lipschitz Properties of Cone-convex Set-valued Functions
• 8. Efficiencies and Optimality Conditions in Vector Optimization with Variable Ordering Structures
• 9. Vectorial Penalization in Multi-objective Optimization
• 10. On Classes of Set Optimization Problems which are Reducible to Vector Optimization Problems and its Impact on Numerical Test Instances
• 11. Abstract Convexity and Solvability Theorems
• 12. Regularization Methods for Scalar and Vector Control Problems
• Index