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This volume presents five surveys with extensive bibliographies and six original contributions on set optimization and its applications in mathematical finance and game theory. The topics range from more conventional approaches that look for minimal/maximal elements with respect to vector orders or set relations, to the new complete-lattice approach that comprises a coherent solution concept for set optimization problems, along with existence results, duality theorems, optimality conditions, variational inequalities and theoretical foundations for algorithms. Modern approaches to scalarization methods can be found as well as a fundamental contribution to conditional analysis. The theory is tailor-made for financial applications, in particular risk evaluation and [super-]hedging for market models with transaction costs, but it also provides a refreshing new perspective on vector optimization. There is no comparable volume on the market, making the book an invaluable resource for researchers working in vector optimization and multi-criteria decision-making, mathematical finance and economics as well as [set-valued] variational analysis.

Inhaltsverzeichnis

Frontmatter

Surveys

Frontmatter

A Comparison of Techniques for Dynamic Multivariate Risk Measures

Abstract
This paper contains an overview of results for dynamic multivariate risk measures. We provide the main results of four different approaches. We will prove under which assumptions results within these approaches coincide, and how properties like primal and dual representation and time consistency in the different approaches compare to each other.
Zachary Feinstein, Birgit Rudloff

Nonlinear Scalarizations of Set Optimization Problems with Set Orderings

Abstract
This paper concerns with scalarization processes of set-valued optimization problems, whose objective space is a Hausdorff locally convex topological linear space and the preferences between the objective values are stated through set orderings. To be precise, general necessary and sufficient optimality conditions for minimal and weak minimal solutions of these optimization problems are obtained by dealing with abstract scalarization mappings that satisfy certain order preserving and order representing properties. Then these conditions are applied to well-known scalarization mappings in set optimization. This approach extends and unifies the main nonlinear scalarization results of the literature on set optimization problems with set orderings.
César Gutiérrez, Bienvenido Jiménez, Vicente Novo

Set Optimization—A Rather Short Introduction

Abstract
Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems.
Andreas H. Hamel, Frank Heyde, Andreas Löhne, Birgit Rudloff, Carola Schrage

A Survey of Set Optimization Problems with Set Solutions

Abstract
This paper presents a state-of-the-art survey on set-valued optimization problems whose solutions are defined by set criteria. It provides a general framework that allows to give an overview about set-valued optimization problems according to decision concepts based on certain set relations. The first part of this paper (Sects. 1 and 2) motivates and describes the set-valued optimization problem (in short, SVOP). The present survey deals with general problems of set-valued optimization and recall its main properties in order to establish the differences between vector set-valued optimization problems (VOP) and set optimization problems (SOP). In this context, in the second part (Sects. 35) we focus on those results existing in the literature related with optimality conditions by using a set approach. We list and quote references devoted to (SOP) from the beginning up to now. In Sect. 5, a particular attention is paid to applications of the set relations considered in other fields as fixed point theory. The last section provides some conclusions and suggestions for further study.
Elvira Hernández

Linear Vector Optimization and European Option Pricing Under Proportional Transaction Costs

Abstract
A method for pricing and superhedging European options under proportional transaction costs based on linear vector optimisation and geometric duality developed by Löhne and Rudloff (Int. J. Theor. Appl. Finance 17(2): 1450012–1–1450012–33, 2014) is compared to a special case of the algorithms for American type derivatives due to Roux and Zastawniak (Acta Applicandae Mathematicae, published online 2015). An equivalence between these two approaches is established by means of a general result linking the support function of the upper image of a linear vector optimisation problem with the lower image of the dual linear optimisation problem.
Alet Roux, Tomasz Zastawniak

Special Topics

Frontmatter

Conditional Analysis on $$\mathbb {R}^d$$

Abstract
This paper provides versions of classical results from linear algebra, real analysis and convex analysis in a free module of finite rank over the ring \(L^0\) of measurable functions on a \(\sigma \)-finite measure space. We study the question whether a submodule is finitely generated and introduce the more general concepts of \(L^0\)-affine sets, \(L^0\)-convex sets, \(L^0\)-convex cones, \(L^0\)-hyperplanes and \(L^0\)-halfspaces. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study \(L^0\)-linear, \(L^0\)-affine, \(L^0\)-convex and \(L^0\)-sublinear functions and introduce notions of continuity, differentiability, directional derivatives and subgradients. We use a conditional version of the Bolzano–Weierstrass theorem to show that conditional Cauchy sequences converge and give conditions under which conditional optimization problems have optimal solutions. We prove results on the separation of \(L^0\)-convex sets by \(L^0\)-hyperplanes and study \(L^0\)-convex conjugate functions. We provide a result on the existence of \(L^0\)-subgradients of \(L^0\)-convex functions, prove a conditional version of the Fenchel–Moreau theorem and study conditional inf-convolutions.
Patrick Cheridito, Michael Kupper, Nicolas Vogelpoth

Set Optimization Meets Variational Inequalities

Abstract
We study necessary and sufficient conditions to attain solutions of set optimization problems in terms of variational inequalities of Stampacchia and Minty type. The notion of solution we deal with has been introduced by Heyde and Löhne in 2011. To define the set-valued variational inequality, we introduce a set-valued directional derivative that we relate to Dini derivatives of a family of scalar problems. Optimality conditions are given by Stampacchia and Minty type variational inequalities, defined both by set-valued directional derivatives and by Dini derivatives of the scalarizations. The main results allow to obtain known variational characterizations for vector optimization problems as special cases.
Giovanni P. Crespi, Carola Schrage

Estimates of Error Bounds for Some Sets of Efficient Solutions of a Set-Valued Optimization Problem

Abstract
In this paper, we establish some estimates of the global/local error bounds for the sets \(S^{\mathrm{Pareto}}_{\bar{y}}\), \(S^{\mathrm{W}}_{\le \bar{y}}\) and \(S^{\mathrm{W}}\), where \(S^{\mathrm{Pareto}}_{\bar{y}}\) is the set of efficient solutions of a unconstrained set-valued optimization problem (\(\mathcal {SP}\)) corresponding to an efficient value \(\bar{y}\) of a unconstrained set-valued optimization problem (\(\mathcal {SP}\)), \(S^{\mathrm{W}}_{\le \bar{y}}\) is the set of weakly efficient solutions of (\(\mathcal {SP}\)) corresponding to weakly efficient values smaller than a weakly efficient value \(\bar{y}\) and \(S^{\mathrm{W}}\) is the set of all weakly efficient solutions of (\(\mathcal {SP}\)). These estimates are expressed in terms of the approximate coderivative, the limiting Fréchet/basic coderivatives and the coderivative of convex analysis. Thus, we establish conditions ensuring the existence of weak sharp minima for (\(\mathcal {SP}\)). We also extend the concept of the good asymptotic behavior to a convex or cone-convex set-valued map.
Truong Xuan Duc Ha

On Supremal and Maximal Sets with Respect to Random Partial Orders

Abstract
The paper deals with definition of supremal sets in a rather general framework where deterministic and random preference relations (preorders) and partial orders are defined by continuous multi-utility representations. It gives a short survey of the approach developed in (J. Math. Econ. 14(4–5):554–563, 2011 [4]), (J. Math. Econ. 49(6):478–487, 2013 [5]) with some new results on maximal sets.
Yuri Kabanov, Emmanuel Lepinette

Generalized Minimality in Set Optimization

Abstract
In this paper, we propose a generalized minimality in set optimization. At first, we introduce parametrized embedding functions, which includes the embedding function in the previous literatures. By using the embedding functions, we generalize notions of minimal solutions for set optimization, and give existence results of the generalized minimal solutions. Also we introduce parametrized scalarizing functions which are generalizations of scalarizing functions defined in the previous literatures, and we characterize the generalized minimal solutions by using the scalarizing functions.
Daishi Kuroiwa

On Characterization of Nash Equilibrium Strategy in Bi-Matrix Games with Set Payoffs

Abstract
In this paper, we consider set-valued payoff bi-matrix games where each player’s payoffs are given by non-empty sets in n-dimensional Euclidean spaces \( \mathbb {R}^{n} \). First, we define several types of set-orderings on the set of all non-empty subsets in \(\mathbb {R}^{n}\). Second, by using these orderings, we define four kinds of concepts of Nash equilibrium strategies to the games and investigate their properties. Finally, we give sufficient conditions for which there exists these types of Nash equilibrium strategy.
Takashi Maeda
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