Operations Research in Progress by Bruno Brosowski (auth.), Gustav Feichtinger, Peter Kall

By Bruno Brosowski (auth.), Gustav Feichtinger, Peter Kall (eds.)

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I,± The random variables Zn,j and ( i v) conditionally independent given Theorem. j. ,. 7n -1 rn =O(c n ) o(b n ) an 00 0 Then under assumption (A). d (X ,S): = in f n SE S (5 ) where S \I Xn - s \I . ) is the (convex) set of solutions of (1). are 52 G. Ch. PFLUG Proof. During the proof we shall use the following notation: By x the projection of x onto the convex set S is denoted. The symbol K is used for an universal constant, which may change its value (similarly to Landau's symbols 0(·) and 0(')), The proof is somewhat lenghty and therefore divided into several steps.

Duality theorems and optimality conditions as esenntial parts of optimization theory are published in [4], [7] , [6) • In the following we give some new results concerning I-conjugate functions and their application to optimization problems in infinite dimensional spaces. Especially duality theorems (including Lagrange duality) and optimality conditions are derived. Other results, which are connected with duality and stability and moreover, with methods for solving nonconvex problems, are only mentioned here.

If -biconjugate function ~-conjugate functions have several interesting properties. Some of them hold in a topological linear space 1 and in the corresponding topological dual space 1'. Then always one of the following condi tions (T) or (T') is fulfilled: (T) N [AJ topol. linear space, symmetric mapping, {~ a BeE) (T') follows from (T) by the substitution of = C {oj, L'. B(E)~ l' by B(E) 1'. 1), where b ~ O. Then (i) fi is convex, f§ is concave. (ii) (c +

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