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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 convey their friendship to Klaus Ritter at the social gathering of his 60th birthday. Be­ reason behind Ritter's many neighbors and his foreign popularity between math­ ematicians, discovering members used to be effortless. in truth, constraints at the measurement of the publication required us to restrict the variety of papers. Klaus Ritter has performed very important paintings in numerous components, specially in var­ ious purposes of linear and nonlinear optimization and likewise in reference to facts and parallel computing. For the latter we need to point out Rit­ ter's improvement of transputer computer undefined. The huge scope of his study is mirrored through the breadth of the contributions during this Festschrift. After numerous years of clinical study within the united states, Klaus Ritter was once 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 college instructing tasks, he has made the task of employing mathematical ways to difficulties of to be centrally important.

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From the facts known to hold in a real-valued setting, it is not surprising that the above theorem has a nonsemistrict counterpart. Theorem 3. Let f: X~~ be a vector-valued function on a convex subset Xr;;;L of a real linear space L. If fis concave with respect to the lexmaxmin order relation i.. , then fis quasiconcave with respect to i... Proof. Assume fto be concave with respect to i... Let x,y E X such that J{x) i;;. J{y). If J{x) i> J{y), we are done since then the semistrict arguments of the proof of the preceding theorem take over.

Problem (6) can be fuLLy reduced to Linear optimization. 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.

K(O) can be made arbitrarily large if only 8 is chosen close enough to a, and y, close enough to 1, always under the condition that py8 = a , where 8E Qn(a,T]), yE Qn(0,1), and Jk(O,a). Analytically speaking, we have the following. }k(O» . }k( a)-f{ 1-y)-1c , where ais predefined and yE Qn(0,1). To ensure that Jk(O, a), note that condition (4) entails (8) /3>0 y< a 18 =? a<8 f3

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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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