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

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 get together of his 60th birthday. Be reason behind Ritter's many acquaintances and his overseas popularity between math ematicians, discovering members was once effortless. actually, constraints at the measurement of the ebook required us to restrict the variety of papers. Klaus Ritter has performed vital paintings in a number of parts, particularly in var ious purposes of linear and nonlinear optimization and in addition in reference to information and parallel computing. For the latter we need to point out Rit ter's improvement of transputer pc undefined. The large scope of his learn is mirrored through the breadth of the contributions during this Festschrift. After a number of years of medical learn within the united states, Klaus Ritter used to be ap pointed as complete professor on the college of Stuttgart. when you consider that then, his identify has turn into inextricably attached with the on a regular basis scheduled meetings on optimization in Oberwolfach. In 1981 he grew to become complete professor of utilized arithmetic and Mathematical records on the Technical college of Mu nich. as well as his collage educating tasks, he has made the job of employing mathematical the way to difficulties of to be centrally important.

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**Sample text**

Fitting the model function by means of nonlinear least squares the unknown mass is determined. Experiments and their evaluation confirmed this proceeding very well. Concavity of the Vector-Valued Functions Occurring in Fuzzy Multiobjective Decision-Making Fred Alois Behringer Munich University of Technology, Dept. of Mathematics Abstract: Werners' approach to fuzzy multiobjective decision-making leads to a crisp substitute problem which is a problem of lexicographically extended maxmin optimization.

X) i> J{y). Assume fto be concave with respect to i... 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{x)]ai[J{y)], by Lemma 2 and Lemma 3 because 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 CoroLLary. In view of the above proof it is immediately clear that concave and semistrictly quasiconcave can be replaced by *-concave and semistrictly *-quasiconcave in the above theorem, where * is any of the following concepts: a (ae:(O, 1) predefined}, rationaLLy (aLL a E Q'"l(O,1», midpoint (a=i).

Thus, we must have ~z) i< ~ LI) and ~z) i< ~~. But this contradicts midpoint quasiconcavity (also a consequence of midpoint concavity). The proof is complete. 0 There was no place in the proof of the above theorem where we actually needed midpoint concavity. The only thing needed was the property of being both midpoint quasiconcave and semistrictly midpoint quasiconcave. ) So we have the following Corollary. If an explicitly midpoint quasiconcave vector-valued function f: X~R" on a convex subset Xof a real linear space is semilocally quasiconcave, then it is semistrictly quasiconcave.