By BANK, GUDDAT, KLATTE, KUMMER, TAMMER

ISBN-10: 3034863284

ISBN-13: 9783034863285

ISBN-10: 3034863306

ISBN-13: 9783034863308

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5) where Lv = {lXV IlX E IR}. 5) also for M. 1 we present it here as a separate theorem because it represents an important special case. i = En, S E {I, ... ~O) such that Yj (X 0) <'~ II, , • s + 1, ... , m , . 3. Let X == 1, ... , s. ) is non-empty for all ). 0. 52 3. Point-to-Set Mappings with Given Structure Proof. 1 (3) 1= {i I gi(X) = MI(A) {l, ... ,m} , = A~ Vx E M(AO)} c {x E fEn I gi(X) ~ Ai Vi E I} =l= 0. Since the functions gi are weakly analytic for i E I, MI(AO) is an affine subspace.

E. 3). If we note that {u E IE" I DSQu = O} = = {u E IE .. I Ou O} holds, the corollary then follows from the theorem. Taking to be vectors of f .. ,). of {l, ... 2. Let A be a metric space and let at: A ~ IE" and bi : A ~ IR be continuous functions on A for all i E {l, ... , m}. Let the point-to-set mapping M: A ~ f .. be defined by M()') = {x E IE .. ), i = 1, .. ) , j = s 1, ... m}, + ). 35) where 1 ~ s ~ m. ) for all Jc {1, ... O) is non-empty and let there exist a b 0 such that V). O). O) V).

Proof. Assume that the inclusion is false. 8) there exist a Z E ri X;. and a k E! ch X;. such that g,,(z, A} = O. 8) one may find ayE X;. , there exists an rx 0 such that < > 3. Point-to-Set Mappings with Given Structure 44 z + ex(z - y) E Xi.. ) 1 +ex and thus to a contradiction. 4). 1. 0E A. O for each I c {I, ... O) is a bounded and non-empty set and =1= I c {I, ... 0. ° ° Proof. 3 respectively. ), }. 1 the relation < O,j E {I, ... ) < O,j E {I, ... ,m}\I}. 2. O). o. o. 6 cannot be evoked.

### Non-Linear Parametric Optimization by BANK, GUDDAT, KLATTE, KUMMER, TAMMER

by Anthony

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