Vector Optimization
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Vector Optimization

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Guang-ya Chen
541, Lecture Notes in Economics and Mathematical Systems
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"This book is devoted to vector or multiple criteria approaches in optimization. Topics covered include: vector optimization, vector variational inequalities, vector variational principles, vector minmax inequalities and vector equilibrium problems. In particular, problems with variable ordering relations and set-valued mappings are treated. The nonlinear scalarization method is extensively used throughout the book to deal with various vector-related problems. The results presented are original and should be interesting to researchers and graduates in applied mathematics and operations research. Readers will benefit from new methods and ideas for handling multiple criteria decision problems. TOC:Introduction and Mathematical Preliminaries.- Vector Optimization Problems.- Vector Variational Inequalities.- Vector Variational Principles.- Vector Minimay Inequalities.- Vector Network Equilibrium Problems"
and Mathematical Preliminaries.- Vector Optimization Problems.- Vector Variational Inequalities.- Vector Variational Principles.- Vector Minimax Inequalities.- Vector Network Equilibrium Problems.
Vector optimization model has found many important applications in decision making problems such as those in economics theory, management science, and engineering design (since the introduction of the Pareto optimal solu­ tion in 1896). Typical examples of vector optimization model include maxi­ mization/minimization of the objective pairs (time, cost), (benefit, cost), and (mean, variance) etc. Many practical equilibrium problems can be formulated as variational in­ equality problems, rather than optimization problems, unless further assump­ tions are imposed. The vector variational inequality was introduced by Gi- nessi (1980). Extensive research on its relations with vector optimization, the existence of a solution and duality theory has been pursued. The fundamental idea of the Ekeland's variational principle is to assign an optimization problem a slightly perturbed one having a unique solution which is at the same time an approximate solution of the original problem. This principle has been an important tool for nonlinear analysis and optimization theory. Along with the development of vector optimization and set-valued optimization, the vector variational principle introduced by Nemeth (1980) has been an interesting topic in the last decade. Fan Ky's minimax theorems and minimax inequalities for real-valued func­ tions have played a key role in optimization theory, game theory and math­ ematical economics. An extension was proposed to vector payoffs was intro­ duced by Blackwell (1955).

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