Get Applied Mathematics and Parallel Computing: Festschrift for PDF

By N. Apostolatos (auth.), Dr. Herbert Fischer, Dr. Bruno Riedmüller, Priv.-Doz. Dr. Stefan Schäffler (eds.)

ISBN-10: 3642997899

ISBN-13: 9783642997891

ISBN-10: 3642997910

ISBN-13: 9783642997914

The authors of this Festschrift ready those papers to honour and show their friendship to Klaus Ritter at the party of his 60th birthday. Be­ explanation for Ritter's many associates and his foreign recognition between math­ ematicians, discovering participants used to be effortless. in reality, constraints at the dimension of the publication required us to restrict the variety of papers. Klaus Ritter has performed vital paintings in quite a few parts, specially in var­ ious purposes of linear and nonlinear optimization and in addition in reference to data and parallel computing. For the latter we need to point out Rit­ ter's improvement of transputer computer undefined. The vast scope of his study is mirrored by means of the breadth of the contributions during this Festschrift. After a number of years of medical study within the united states, Klaus Ritter used to be ap­ pointed as complete professor on the collage of Stuttgart. because then, his identify has develop into inextricably hooked up with the frequently scheduled meetings on optimization in Oberwolfach. In 1981 he turned complete professor of utilized arithmetic and Mathematical records on the Technical collage of Mu­ nich. as well as his collage instructing tasks, he has made the task of utilizing mathematical how to difficulties of to be centrally important.

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Here, all concavity concepts are taken with respect to i" . 49 6. LocaL versus gLobaL optimaLity and related remarks There are three well-known results which immediately enter one's mind if quasi concavity or semistrict quasiconcavity is mentioned in connection with optimization: (1) The feasible set defined by a set of inequalities of the form g(X) ;;. 0, where the components ,91(-) are quasiconcave, is convex. (2) The set of optimal points in a (scalar) maximization problem is convex if the objective function is quasiconcave.

To complete the proof, suppose that J{x) i- J{y). ) Choose any aE(O,1}. Then, i[J{x)] l= i[J{y)], by definition because J{x) i- J{y); i[J{xay)] l;;. i[(J{x)aJ{y)], because fis concave; i[J{x)aJ{y)] l;;. i[J{x)]ai[J{y)], by (2) and (3) of Lemma 1; i[J{x)]ai[J{y)] l= i[J{y)]ai[J{y)], since i[J{x)] l= i[J{y)]; i[J{y)]ai[J{y)] l= i[J{y)], which is immediate. Hence, i[J{xay)] l;;. i[J{y)], which means that J{xay) i;;. J{Y), by definition. The proof is complete. 0 46 Corollary. In view of the above proof it is immediately clear that concave and quasi concave can be replaced by *-concave and *-quasiconcave in the above theorem, where * is any of the folLoVlling concepts: a (ae(O, 1) predefined), rationally (all a E (0,1 )nQ) , midpoint (a=~) .

Moreover, there stiLL remains the possibility of reducing problem (6) to a set of consecutive maxmin problems if the individual functions are quasi concave 41 and semistrictLy quasiconcave. JL(-» := (h1(,u1(-»'··· ,~(,uk(-)W is used in probLem (6) instead of 11(-), 'Nhere the hi'S are strictLy monotone increasing. S-shaped functions do have this property. PracticaL experiences reported by LeberLing [19], Zimmermann and Zysno [32], and others (see Werners [30)) show that a decision-maker's behaviour towards fuzzy probLems usuaLLy is of such an "S-shaped" nature.

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Applied Mathematics and Parallel Computing: Festschrift for Klaus Ritter by N. Apostolatos (auth.), Dr. Herbert Fischer, Dr. Bruno Riedmüller, Priv.-Doz. Dr. Stefan Schäffler (eds.)

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