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By Richard B. Holmes (auth.)

ISBN-10: 3540057641

ISBN-13: 9783540057642

ISBN-10: 3540371826

ISBN-13: 9783540371823

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Convex Programs a) D e f i n i t i o n . where X is a set and function. matical value A variational The associated program, When X problem All is a is and is called set which (K,f), is not where K (X), in X (X,f) the o b j e c t i v e or a b s t r a c t inf f(X), (if any) the a s s o c i a t e d mathe- called where solutions at the outset of m i n i m i z i n g space. is a convex problem, pair the the value of the program. variational c o n v e x ~rogram. to r e c o g n i z e a linear is c a l l e d are then c a l l e d f s Conv the p r o b l e m f the n u m b e r and the points an a b s t r a c t It is i m p o r t a n t encompasses variational such points is an o r d e r e d (-~,+~]; is to d e t e r m i n e of the p r o g r a m , is attained.

The basic result, which depends on the Theorem in d), is the following. Theorem, as in 12c). ,fn For given inf {f(-): define an ordinary Y, F ~ Rn let x, ~ e X fi(. ) <_ yi }, resp. ) < ~i }. Lagrange multiplier vectors, If then ~"(y-F) ! f ( x ) - f ( x ) ! i - . ( y - y ) . Proof. ,fn-y- n. perturbation p(y) = q(y-~) function the is handled similarly. for the ordinary Then by d)~ Let convex program de- -~- E aq(@). If p is the for the original program, we see that and hence that aq(@) = ap(7).

Ko @ [21]. be a convex set in a real Ics X. 52 if and only if ~ Yi s K~ (23 Yo + Yl +'''+ Yn = @" Proof. f-~K n + ~. ,n. ko < 0 This implies for which Hence • - of -- and e i=O follows that by (3), so Halkin of Ji' ~ 0 where sup Ko,Y ° Yo ~ K°o J is a by 15d). + an = I, Thus ~ Xi = 0. ,n; x ~ K, -< - Z 1(~i + (x,Yi~) of this theorem have been given by Vlach from Ioffe-Tikhomirov so in it -< 0 also, qed. [23], and Pshenichnii variational ), Yi ~ -Xoai J° ~ Xi JO" From this, and the fact that 1 be seen shortly, and Since = co (J U .

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A Course on Optimization and Best Approximation by Richard B. Holmes (auth.)


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