By Richard B. Holmes (auth.)

ISBN-10: 3540057641

ISBN-13: 9783540057642

ISBN-10: 3540371826

ISBN-13: 9783540371823

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B) When then xo Vf(xo) X is a Ics and is a s o l u t i o n = 8. More f a smooth of the p r o g r a m generally, for any convex (X, f) f s Cony function on if and only (X), x° X, if is a s o l u t i o n if and only if (i) e ~ ~f(Xo). Although trivial for later more in itself, informative the o p t i m a l i t y condition characterizations (i) is the basis of s o l u t i o n s of c o n v e x programs. Note x o s dom that, by 9a - R e m a r k f ' ( X o ; X ) >_ o, This condition nbhds, hence of x o.

Kuhn-Tucke r Theory As a second illustration of the use of lld), we consider a special class of convex programs in the finite dimensional ("ordinary convex programs") case, have been of great practical and for which an elegant theory is available. intuitively described as "minimizing which, interest, The programs may be a convex function subject to convex constraints". a) Lemma. ,K n be closed convex bodies whose interiors have a point in common. N(Xo,K) = Let in a ics x ° s K ~ ( ~ K i. X Then ~N(Xo,Ki).

Of the form of the results in a) x ° s ~ K i. ,fn s Cony on K i = {x c X: fi(x) ! 0}. '',n. Such for the last two decades, and duality theory has been develope& [20, 30, 44, SS, 70]. the solutions are continuous Our only concern of these programs under the 34 regularity assumption a point in value. X (or "constraint qualification") where all the fi's simultaneously The existence question for solutions convex programs is frequently discussed here. assume a negative of (ordinary or abstract) easier to answer~ and will not be But see §30 below for some special [70, ~27] for the general d) Theorem.

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

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