Vector Optimization: Theory, Applications, and Extensions by Johannes Jahn

By Johannes Jahn

basics and demanding result of vector optimization in a common environment are provided during this publication. the idea constructed comprises scalarization, life theorems, a generalized Lagrange multiplier rule and duality effects. purposes to vector approximation, cooperative online game idea and multiobjective optimization are defined. the speculation is prolonged to set optimization with specific emphasis on contingent epiderivatives, subgradients and optimality stipulations. historical past fabric of convex research being invaluable is concisely summarized on the beginning.

This moment version includes new elements at the adaptive Eichfelder-Polak technique, a concrete software to magnetic resonance structures in clinical engineering and extra feedback at the contribution of F.Y. Edgeworth and V. Pareto. The bibliography is up to date and comprises newer very important publications.

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For a proof of this lemma see, for instance, Holmes [125, p. 72]. 40. p EX* is surjective. Every reflexive real normed space is complete and, therefore, it is a real Banach space. For the applications the following assertion is important (see Holmes [125, p. 126/127]). 28 Chapter 1. 41. A real Banach space (X, II · II) is reflexive if and only if the closed unit ball { x E X I II · II : : ; 1} is weakly compact. If in a topological linear space a partial ordering is given additionally, it is important to know the relationships between the topology and the ordering.

A) In a real Banach space X an ordering cone Cx is normal for the norm topology if and only if the dual cone Cx· is reproducing. 3. Topological Linear Spaces 29 (b) In a real locally convex space X an ordering cone Cx is normal for the weak topology a(X, X*) if and only if the dual cone Cx· is reproducing. (c) In a real locally convex space X a normal ordering cone is also normal for the weak topology a(X, X*). (d) In a real locally convex space an ordering cone with a bounded base is normal. Order intervals play an important role for the definition of a norm in a real linear space.

5. Let X and Y be real linear spaces, let Cy be a convex cone in Y, letS be a nonempty subset of X, and let f: S-+ Y be a given map. 1. 2: Non-convex functional. 3: Concave functional. is called the epigraph off (see Fig. 4). 5) can also be written as epi(f) = {(x, y) I xES, f(x) :::::cy y}. It turns out that a convex map can be characterized by its epigraph. 6. Let X and Y be real linear spaces, let Cy be a convex cone in Y, let S be a nonempty subset of X and let f : S -t Y be a given map. Then f is convex if and only if epi(f) is a convex set.

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