By Anatoly Antipin (auth.), Prof. Van Hien Nguyen, Prof. Jean-Jacques Strodiot, Dr. Patricia Tossings (eds.)
The ninth Belgian-French-German convention on Optimization has been held in Namur (Belgium) on September 7-11, 1998. This quantity is a suite of papers offered at this convention. initially, this convention used to be a French-German convention yet this 12 months, in line with the organizers' needs, a 3rd kingdom, Belgium, has joined the founding contributors of the convention. accordingly the identify: Belgian French-German convention on Optimization. because the very starting, the aim of those meetings has been to assemble researchers operating within the sector of Optimization and partic ularly to inspire younger researchers to give their paintings. lots of the members come from the organizing nations. but the normal ten dancy is to ask outdoor researchers to wait the assembly. So this yr, one of the a hundred and one contributors at this convention, twenty researchers got here from different nations. the overall topic of the convention is every thing that issues the world of Optimization with out specification of specific issues. So theoretical as pects of Optimization, as well as purposes and algorithms of Opti mization, should be built. notwithstanding, and this element used to be extremely important for the organizers, the convention needs to continue its convivial personality. not more than parallel periods are prepared. this might let helpful contacts among researchers to be promoted. The editors convey their honest due to all those that took half during this convention. Their helpful discussions have made this quantity possible.
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Extra resources for Optimization: Proceedings of the 9th Belgian-French-German Conference on Optimization Namur, September 7–11, 1998
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This result is wellknown (see for instance [1]). The differential inclusion appearing as the last condition in (SC) is not amenable for numerical realization and we therefore replace it by ). * =c (* u ). * IIU +-c a d (* U ). * )) +C ). * b) , =cmax (0 ,u • +-C (1) for any c > O. Here IIuad denotes the Hilbert space projection of L2(il) onto Uad and max stands for the pointwise maximum as x varies in il. e. in Q . "itine Bergounioux and Karl Kunisch In this very case we assume Ud E H2(D) and 'P E H 4 (D).
190703e-02 4. 190712e-02 4. 190712e-02 4. 190712e-02 4. 190785e-02 4. 190712e-02 4. 190712e-02 4. a. •• Fig. 1. Optimal State and Optimal control We present in Table 4 a second example where (8) may be not fulfilled (it was for Table 3) because 0: is too small; in addition Ud has been chosen infeasible. Though the size of the set An, in the sense of number of grid points in is increasing, the sequence An does not increase monotonically. More precisely points in An at iteration n may not belong to An+!
L ::; t. Then TI-£ is a contraction from Proof of lemma 1. l ::; the operator v t-;. lV¢(v) is a contraction. l(V¢(v) - V¢(w),v - w) In order to prove that v to show that t-;. l2IV¢(v) - V¢(wW. em 2(l that. l. -Lipsehitz property of \7p, see lemma IL 2 below. So, for 0 < fl :::; f' we obtain that TIL is a eontradion. 0 Lemma 2 (Baillon-Haddad [5]). For an opemtor- \7p (the gmdient of a convex function), the following pmper-ties (i) and (ii) aTe equivalent (i) 'iv,w E H, 1\7¢(v) - \7¢(w) I :::; Llv - 101, (ii) 'iv,w E H, (\7(v) - \7(10),11 - 10) 2: il\7¢(v) - \7(wW.