By Asen L. Dontchev, Tullio Zolezzi (auth.)

ISBN-10: 354047644X

ISBN-13: 9783540476443

ISBN-10: 3540567372

ISBN-13: 9783540567370

This e-book provides in a unified manner the mathematical concept of well-posedness in optimization. the elemental innovations of well-posedness and the hyperlinks between them are studied, particularly Hadamard and Tykhonov well-posedness. summary optimization difficulties in addition to purposes to optimum keep an eye on, calculus of diversifications and mathematical programming are thought of. either the natural and utilized part of those themes are awarded. the most topic is frequently brought by way of heuristics, specific circumstances and examples. whole proofs are supplied. the predicted wisdom of the reader doesn't expand past textbook (real and practical) research, a few topology and differential equations and simple optimization. References are supplied for extra complicated issues. The publication is addressed to mathematicians attracted to optimization and comparable themes, and likewise to engineers, keep watch over theorists, economists and utilized scientists who can locate right here a mathematical justification of sensible approaches they encounter.

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Suppose (K, I ) fails to be Levitin - Polyak well-posed. Then there exists a generalized minimizing sequence Un such that ]un] _> a > 0 for some a E (0, 1) and every n. If un is bounded it has some cluster point y, with y E arg min (K, I) by lower semicontinuity and dist ( u n , K ) ~ 0. Hence I ( y ) = 0, contradicting well-posedness since ]y] > a. If ]un[ ---* +cx), by convexity of dist (-, K ) we get 'a n dist ( ~ - ~ , A) ---* 0 and by convexity of I we see that un/]un ] is a bounded generalized minimizing sequence.

Notes and Bibliographical Remarks. To section 1. The fundamental definition of Tykhonov well-posedness goes back to Tykhonov [5]. Example 7 may be found in Rockafellar [2, p. 266] where (section 27) further well-posedness criteria can be found. Remark 9 is due to Porack~ - Divig [1]. To section 2. The basic theorem 11 is due to Furi - Vignoli [3]. Theorem 12 and proposition 15 are due to Zolezzi [8], extending previous results of Vajnberg [1], where further sufficient conditions to well-posedness may be found.

4 we see that the conclusion of theorem 4 fails if B is replaced by a bounded closed convex set. 42 PROOFS. 5 LemmaLetcER, uEX*,u¢0, H = { y E X :< u , y > = c } . T h e n for every x E X dist (x, H ) = l < u, x > -cl/ll~ll. P r o o f . If y E H I < u, x > - c l ~ I < u, • - y > I ~ Ilull IIx - ytl giving dist (x, H ) >_ l < u, x > -cl/llull. We show now the converse inequality. Given e > 0, there exists z E X such t h a t Ilzll = 1 a n d I l u l l - c < < u, z > . T h e n y=x U, X :> - - C < tt, Z > zEH, moreover IIx - yll = II < u , x > - C z l I < I < u,x > -el Ilull-~ ' so by letting s --~ 0 dist (x, H ) < I < u, z > -cl/llull.

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Well-Posed Optimization Problems by Asen L. Dontchev, Tullio Zolezzi (auth.)


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