Multiple Criteria Optimization: State of the Art Annotated by Matthias Ehrgott (Editor), Xavier Gandibleux (Editor)
By Matthias Ehrgott (Editor), Xavier Gandibleux (Editor)
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Extra info for Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys (International Series in Operations Research & Management Science)
We show, that is closely related to a multiparametric vector optimization problem which we call the corresponding Dinkelbachtransformed problem, namely where i = 1, …, m and is a parameter which must be chosen in a suitable way. The original result of Dinkelbach  from 1967 (and also the foregoing result of Jagannathan  from 1966 for linear fractional problems) concerns the case m = 1 with only one objective function and says that a given point is optimal for iff it is optimal for with Corresponding results for the sets of efficient and properly efficient solutions, respectively, of both problems in the case were given by Bector and Chandra , Kaul and Lyall , Weir  and others.
This leads to converse duality theorems. 28), for all there is an such that 28 MULTIPLE CRITERIA OPTIMIZATION Theorem 13 (Converse duality theorem) Assume that both (V1) and (V2) hold and Then there are Y with such that The duality in the last section is an abstract and nonscalarized Lagrangean formalism for very general optimization problems. 3 we will present pairs of dual problems for special classes of vector optimization problems. 2. Duality by Scalarization Consider (P) in the following form with where is a nonempty set in V and X, Y, V are topological vector spaces, Y Hausdorff, Y partially ordered by C closed and with nonempty interior.
1 in  the function belongs to the class and for each it holds and, hence, with Hence, for the special case m = 1 (in which we can put without loss of generality µ = 1) the function G belongs even to the class with In this way the iteration rule of the Newton-method has the very simple form which is nothing else than the iteration rule of Dinkelbach , who used this rule also under other assumptions as given in (A), since convergence results are very much easier to obtain in the one-dimensional case.