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.
Table of Content
Introduction.- Sets and Binary Relations.- Extended Real-Valued Functions.- Translation Invariant Functions.- Minimizers of Translation Invariant Functions.- Vector Optimization in General Spaces.- Multiobjective Optimization.- Variational Analysis.- Special Cases and Functionals Related to φA, k.- Set-Valued Optimization Problems.- Vector Optimization With Variable Domination Structures.- Variational Methods in Topological Vector Spaces.- Algorithms for the Solution of Optimization Problems.- Optimization Under Uncertainty.- Further Applications.
About the author
Christiane Tammer is Professor at Martin-Luther-University Halle-Wittenberg in Halle (Saale), Gemany. She is working in the field of variational analysis and optimization. She has co-authored four monographs: 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 Editor in Chief of the journal Optimization, Co-Editor in Chief of the journal Applied Set-Valued Analysis and 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.Petra Weidner is Professor for Mathematics and Computer Science at HAWK University of Applied Sciences and Arts Hildesheim/Holzminden/Göttingen in Göttingen, Germany. She has worked at Martin-Luther-University Halle-Wittenberg and University Hamburg as well as for a business consulting firm. Her research interests include fundamentals of nonlinear functional analysis, operations research, vector optimization and decision making as well as solution methods for multiobjective optimization problems.
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