This book presents fundamentals and important results of vector optimization in a general setting. The theory developed includes scalarization, existence theorems, a generalized Lagrange multiplier rule and duality results. Applications to vector approximation, cooperative game theory and multiobjective optimization are described. The theory is extended to set optimization with particular emphasis on contingent epiderivatives, subgradients and optimality conditions. Background material of convex analysis being necessary is concisely summarized at the beginning. This second edition contains new parts on the adaptive Eichfelder-Polak method, a concrete application to magnetic resonance systems in medical engineering and additional remarks on the contribution of F.Y. Edgeworth and V. Pareto. The bibliography is updated and includes more recent important publications. In vector optimization one investigates optimal elements such as min imal, strongly minimal, properly minimal or weakly minimal elements of a nonempty subset of a partially ordered linear space. The prob lem of determining at least one of these optimal elements, if they exist at all, is also called a vector optimization problem. Problems of this type can be found not only in mathematics but also in engineer ing and economics. Vector optimization problems arise, for exam ple, in functional analysis (the Hahn-Banach theorem, the lemma of Bishop-Phelps, Ekeland's variational principle), multiobjective pro gramming, multi-criteria decision making, statistics (Bayes solutions, theory of tests, minimal covariance matrices), approximation theory (location theory, simultaneous approximation, solution of boundary value problems) and cooperative game theory (cooperative n player differential games and, as a special case, optimal control problems). In the last decade vector optimization has been extended to problems with set-valued maps. This new field of research, called set optimiza tion, seems to have important applications to variational inequalities and optimization problems with multivalued data. The roots of vector optimization go back to F. Y. Edgeworth (1881) and V. Pareto (1896) who has already given the definition of the standard optimality concept in multiobjective optimization. But in mathematics this branch of optimization has started with the leg endary paper of H. W. Kuhn and A. W. Tucker (1951). Since about v Vl Preface the end of the 60's research is intensively made in vector optimization. Front Matter....Pages i-xv Front Matter....Pages 1-2 Linear Spaces....Pages 3-36 Maps on Linear Spaces....Pages 37-59 Some Fundamental Theorems....Pages 61-100 Front Matter....Pages 101-102 Optimality Notions....Pages 103-114 Scalarization....Pages 115-148 Existence Theorems....Pages 149-160 Generalized Lagrange Multiplier Rule....Pages 161-188 Duality....Pages 189-207 Front Matter....Pages 209-210 Vector Approximation....Pages 211-242 Cooperative n Player Differential Games....Pages 243-278 Front Matter....Pages 279-280 Theoretical Basics of Multiobjective Optimization....Pages 281-313 Numerical Methods....Pages 315-349 Multiobjective Design Problems....Pages 351-381 Front Matter....Pages 383-384 Basic Concepts and Results of Set Optimization....Pages 385-392 Contingent Epiderivatives....Pages 393-409 Subdifferential....Pages 411-421 Optimality Conditions....Pages 423-447 Back Matter....Pages 449-481 Fundamentals And Important Results Of Vector Optimization In A General Setting Are Presented In This Book. The Theory Developed Includes Scalarization, Existence Theorems, A Generalized Lagrange Multiplier Rule And Duality Results. Applications To Vector Approximation, Cooperative Game Theory And Multiobjective Optimization Are Described. The Theory Is Extended To Set Optimization With Particular Emphasis On Contingent Epiderivatives, Subgradients And Optimality Conditions. Background Material Of Convex Analysis Being Necessary Is Concisely Summarized At The Beginning. This Second Edition Contains New Parts On The Adaptive Eichfelder-polak Method, A Concrete Application To Magnetic Resonance Systems In Medical Engineering And Additional Remarks On The Contribution Of F.y. Edgeworth And V. Pareto. The Bibliography Is Updated And Includes More Recent Important Publications.
this Book Presents Fundamentals And Important Results Of Vector Optimization In A General Setting. The Theory Developed Includes Scalarization, Existence Theorems, A Generalized Lagrange Multiplier Rule And Duality Results. Applications To Vector Approximation, Cooperative Game Theory And Multiobjective Optimization Are Described. The Theory Is Extended To Set Optimization With Particular Emphasis On Contingent Epiderivatives, Subgradients And Optimality Conditions. Background Material Of Convex Analysis Being Necessary Is Concisely Summarized At The Beginning.
