In vector optimization one investigates optimal elements such as minยญ imal, strongly minimal, properly minimal or weakly minimal elements of a nonempty subset of a partially ordered linear space. The probยญ 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 engineerยญ ing and economics. Vector optimization problems arise, for examยญ ple, in functional analysis (the Hahn-Banach theorem, the lemma of Bishop-Phelps, Ekeland's variational principle), multiobjective proยญ 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 optimizaยญ 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 legยญ 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
CONTENT
I Convex Analysis -- 1 Linear Spaces -- 2 Maps on Linear Spaces -- 3 Some Fundamental Theorems -- II Theory of Vector Optimization -- 4 Optimality Notions -- 5 Scalarization -- 6 Existence Theorems -- 7 Generalized Lagrange Multiplier Rule -- 8 Duality -- III Mathematical Applications -- 9 Vector Approximation -- 10 Cooperative n Player Differential Games -- IV Engineering Applications -- 11 Theoretical Basics of Multiobjective Optimization -- 12 Numerical Methods -- 13 Multiobjective Design Problems -- V Extensions to Set Optimization -- 14 Basic Concepts and Results of Set Optimization -- 15 Contingent Epiderivatives -- 16 Subdifferential -- 17 Optimality Conditions -- List of Symbols
