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Like norms, translation invariant functions are a natural and powerful tool for the separation of sets and scalarization. This book provides an extensive foundation for their application. It presents in a unified way new results as well as results which are scattered throughout the literature. The functions are defined on linear spaces and can be applied to nonconvex problems. Fundamental theorems for the function class are proved, with implications for arbitrary extended real-valued functions. The scope of applications is illustrated by chapters related to vector optimization, set-valued optimization, and optimization under uncertainty, by fundamental statements in nonlinear functional analysis and by examples from mathematical finance as well as from consumer and production theory.

The book is written for students and researchers in mathematics and mathematical economics. Engineers and researchers from other disciplines can benefit from the applications, for example from scalarization methods for multiobjective optimization and optimal control problems.

### Chapter 1. Introduction

Abstract
The introduction describes the outline of the book. It starts with the definition and the formula for translation invariant functions. The scope of applications is indicated. In the introduction, the interdependencies between different chapters and sections are explained. This is of relevance since general approaches to theoretical and applied problems are deduced from the presented foundations. Moreover, Petra Weidner is the author of Chapters 2–7 and Christiane Tammer is the author of Chaps. 8–15.
Christiane Tammer, Petra Weidner

### Chapter 2. Sets and Binary Relations

Abstract
In the first sections of this chapter, we provide well-known fundamentals of linear spaces and topological spaces, binary relations and cones. We add some new results and details that are necessary for the understanding of the following chapters and for the proofs therein. Furthermore, our notation is introduced. We investigate properties of recession cones and norms which are defined by means of cones. Statements about the algebraic interior and the algebraic closedness of sets are summarized. We introduce the directional closedness of sets in linear spaces together with related notions and prove direction-depending properties of sets. These properties build the basis for a full characterization of translation invariant functions. They allow to deduce results in cases where usually one of the following conditions has to be fulfilled, but is not satisfied or too restrictive:
- the space is equipped with a topology,
- a set is closed or has a nonempty interior in a given topology,
- a set is algebraically closed or its algebraic interior is nonempty.
Petra Weidner

### Chapter 3. Extended Real-Valued Functions

Abstract
In this chapter, we present basic notations and properties for functions which attain values in $$\mathbb {R}\cup \{-\infty ,+\infty \}$$, the extended set of real numbers. Of course, such functions may also be real-valued. In the first two sections, we give a short overview about the extended set of real numbers and explain the way in which extended real-valued functions are handled in convex analysis. We also discuss problems that arise by admitting values $$\pm \infty$$. Afterwards, statements about the continuity and the semicontinuity of extended real-valued functions are summarized. We investigate convexity, generalized convexity, linearity, sublinearity and related properties of the functionals. One section is devoted to monotonicity. Furthermore, basic notions for the variational analysis of extended real-valued functions are collected. Since extended real-valued functions belong to the standard in convex analysis, most of the results in this chapter are well known. We add several new statements in order to point out that some properties of the functions which we focus on in this book turn out to be properties of each extended real-valued function.
Petra Weidner

### Chapter 4. Translation Invariant Functions

Abstract
This chapter contains the foundations for the application of translation invariant functions. Fundamental properties of translation invariant functions and separation theorems for not necessarily convex sets in linear spaces are proved. Ways to construct translation invariant functions in general as well as with desired properties are discussed in detail. Semicontinuity, continuity, convexity, concavity and monotonicity belong to these properties. Monotonicity is investigated in the framework of scalarizing binary relations. Special sections refer to translation invariant functions generated by linear inequalities and to the variational analysis for translation invariant functions. The approximimation of translation invariant functions by convex or concave functions is examined. Relationships are proved between translation invariant functions on the one hand, and Minkowski functionals and norms on the other hand. Moreover, we derive formulas for translation invariant extensions as well as for translation invariant envelopes of arbitrary extended real-valued functions. In this way, a tool for the investigation of arbitrary extended real-valued functions is obtained. We illustrate its use by describing continuity via properties of the epigraph and by a formula for the inf-convolution’s epigraph. Beside this, statements for monotone, lower semicontinuous, and convex envelopes are deduced. Assumptions, definitions and statements related to translation invariant functions have immediate practical interpretations in mathematical economics, especially in mathematical finance as well as in consumer and production theory. Such interpretations are given throughout the chapter. Furthermore, applications in functional analysis and vector optimization will be indicated. Detailed bibliographical notes related to the history of the functions are included.
Petra Weidner

### Chapter 5. Minimizers of Translation Invariant Functions

Abstract
When applying a translation invariant function for the separation of sets or the scalarization of problems, then it usually has to be minimized over a given set. In the subsequent chapters, we will characterize solutions to vector optimization problems by minimizers of translation invariant functions on a feasible set. Good-deal bounds of cash streams turn out to be infima of translation invariant functions. In this chapter, we study minimization problems having a translation invariant objective function. Such problems are connected with a generalization of an optimization problem introduced by Pascoletti and Serafini. The existence of optimal solutions and properties of the solution set are investigated. Relationships to problems with an altered feasible set are proved. The considered problems depend on certain parameters when they are applied to scalarization. We examine the related parameter control. Assumptions are given under which the considered problems are equivalent to the minimization of special Minkowski functionals and norms.
Petra Weidner

### Chapter 6. Vector Optimization in General Spaces

Abstract
In this chapter, vector optimization problems in linear spaces are studied. Here the notion of vector optimization is used in a very general way. A vector optimization problem with a variable domination structure is equivalent to the problem of finding nondominated solutions in a set related to some arbitrary binary relation. Each decision problem can be shown to be of this type. We introduce the basic concept for variable domination structures and connect it to decision problems. Optimal decisions are investigated as optima w.r.t. (with regard to) the preference relation, and the results can straightforwardly be transferred to optima w.r.t. arbitrary binary relations. It will be proved that—even for variable domination structures—efficient points can be helpful in finding the best solutions. Later on, we investigate the case where the domination structure can be described by a single set, namely the domination set of the vector optimization problem. We define efficiency w.r.t. reference sets which are not necessarily domination sets and give a thorough motivation for this approach. Properties of such efficient point sets are investigated. We study efficient and weakly efficient solutions of vector optimization problems, including surrogates for the weakly efficient point set and problems with uncertainties or perturbations. Scalarization results are proved, with an emphasis on translation invariant functions and implications for norms. Beside this, some basic properties of efficient and weakly efficient solutions are examined, especially their existence. The results imply statements for properly efficient solutions and related subsets of the efficient point set.
Petra Weidner

### Chapter 7. Multiobjective Optimization

Abstract
This chapter focuses on scalarization methods in multiobjective optimization and on properties of Geoffrion’s properly efficient point set. Geoffrion’s properly efficient point set is described by efficient and weakly efficient point sets related to cones, especially to different types of polyhedral cones. Moreover, existence results and the density in the Pareto optimal point set are proved under rather mild assumptions. The properly efficient point set is characterized by minimizers of strictly monotone functionals and by minimizers of translation invariant functions. Conditions for the coincidence of Geoffrion’s proper efficiency with Nehse–Iwanow’s proper efficiency are given. Statements which can be deduced from the previous chapters and from the investigation of Geoffrion’s proper efficiency are applied to scalarization procedures in multiobjective optimization. Beside relationships between the solution sets of the scalarizing problems and optimal point sets of the multiobjective optimization problem, the results for the scalar problems include statements about the existence and uniqueness of their solutions as well as about the parameter control. The weighted Chebyshev norm minimization and its extension by Choo and Atkins, Wierzbicki’s reference point projection, the $$\varepsilon$$-constraint method, and the Hurwicz Rule for decision making under uncertainty belong to the examined scalarizations. The results are relevant to many other scalarization methods since a general framework for the systematic investigation of such methods is presented. This framework is based on translation invariant functions.
Petra Weidner

### Chapter 8. Variational Analysis

Abstract
In this chapter, we recall well-known concepts of generalized differentiation for vector-valued functions as well as set-valued maps. These notions are important for deriving optimality conditions. Here the focus is on developing differentiability properties of nonconvex functions (maps).
Christiane Tammer

### Chapter 9. Special Cases and Functionals Related to

Abstract
We study translation invariant functionals where the involved set is constructed by block norms or oblique norms. These sets are described by linear inequalities. We characterize efficient elements by means of functionals where oblique norms are involved and weakly efficient elements via scalarization using block norms. Our results are applied to multiobjective d.c. (difference of convex functions) optimization problems. Furthermore, we discuss several fundamental properties of the directional minimal time function that are important for applications in locational analysis. We present relationships between the nonlinear translation invariant functionals and the oriented distance by Hiriart-Urruty.
Christiane Tammer

### Chapter 10. Set-Valued Optimization Problems

Abstract
Set optimization has become a very important field in optimization theory as well as in various applications, especially in welfare economics, mathematical finance, optimization under uncertainty and medical image processing. The aim of this chapter is to show that it is possible to give characterizations of solutions of set-valued optimization problems by means of translation invariant functionals. First, we introduce set relations and define solution concepts for set-valued optimization problems based on these set relations. Hernández and Rodríguez-Marín introduced a functional that can be considered as an extension of translation invariant functionals related to set-valued optimization. We discuss the properties of this functional, especially boundedness and monotonicity. A characterization of the solutions of set-valued optimization problems via scalarization by means of translation invariant functionals is given. We will see that it is possible to derive a complete characterization of solutions to set-valued optimization problems using translation invariant functionals with very useful properties and nice geometrical interpretations.
Christiane Tammer

### Chapter 11. Vector Optimization with Variable Domination Structures

Abstract
Vector optimization with variable domination structures is a growing up and expanding field of applied mathematics that deals with optimization problems where the domination structure is given by a set-valued map acting between abstract or finite-dimensional spaces. Interesting and important applications of vector optimization with variable domination structure arise in economics, psychology, capability of human behavior, in portfolio management, location theory and radiotherapy treatment in medicine. We give a detailed discussion of solution concepts for problems with variable domination structures based on the (pre-) domination structures. We present certain modifications of translation invariant functionals and show characterizations of solutions to vector optimization problems with variable domination structure by means of translation invariant functionals as well as corresponding modifications. Furthermore, we introduce several concepts for approximate solutions to vector optimization problems with variable domination structures and show corresponding characterizations by means of translation invariant functionals. These results are very useful for further research in the field of vector optimization with variable domination structure, especially for deriving optimality conditions, duality assertions and numerical procedures.
Christiane Tammer

### Chapter 12. Variational Methods in Topological Vector Spaces

Abstract
In this chapter, we explain that the application of translation invariant functionals (or its modifications) is an important tool for deriving necessary optimality conditions in vector optimization, duality assertions, minimal point theorems and variational principles, necessary conditions for approximate solutions of vector optimization problems with respect to variable domination structure, existence results for solutions of vector variational inequalities and many other results in optimization theory and functional analysis. Furthermore, we derive certain relationships to the Extremal Principle by Kruger and Mordukhovich. For the proofs of these results, the separation properties of translation invariant functionals as well as the monotonicity, convexity and continuity properties are essential.
Christiane Tammer

### Chapter 13. Algorithms for the Solution of Optimization Problems

Abstract
For generating solutions to vector optimization problems via algorithms based on a scalarization, the monotonicity properties of the scalarizing functionals are important. In this chapter, we present Benson’s Outer Approximation Algorithm that uses a scalarization by means of translation invariant functionals. Furthermore, we present proximal-point algorithms as well as an adaptive algorithm for solving vector optimization problems where translation invariant functionals are involved. We show that a scalarization by means of translation invariant functionals is useful for deriving an algorithm for solving set-valued optimization problems. Finally, we derive algorithms for solving vector optimization problems with variable domination structure using an extension of translation invariant functionals.
Christiane Tammer

### Chapter 14. Optimization Under Uncertainty

Abstract
Robust optimization is a very active field of research. In this chapter, we show that translation invariant functionals can be considered in order to describe many concepts of robustness and stochastic optimization which are well known from scalar optimization under uncertainty as special cases of a scalarization by means of translation invariant functionals. Based on this unified approach to robustness and stochastic optimization, it is possible to derive new concepts of robustness in scalar optimization under uncertainty. Moreover, we explain that the well-studied properties of translation invariant functionals allow the establishment of useful relationships to multiobjective optimization problems.
Christiane Tammer

### Chapter 15. Further Applications

Abstract
In this chapter, we show that coherent risk measures in mathematical finance can be formulated using translation invariant functionals such that it is possible to use the results proved in this book in order to derive corresponding properties for coherent risk measures. Furthermore, we study the relationship between coherent risk measures and a strictly robust counterpart problem of an optimization problem under uncertainty. The benefit function and shortage function in mathematical economics are related to translation invariant functionals. Moreover, we consider a vector-valued optimal control problem with PDE-constraints and apply the scalarization technique by means of translation invariant functionals for deriving characterizations of solutions to this vector-valued optimal control problem that are useful for corresponding adaptive algorithms. Finally, we use the directional minimal time function for the formulation of location problems and present necessary optimality conditions for solutions of these (in general nonconvex) location problems.
Christiane Tammer