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Download Nonlinear Programming 2. Proceedings of the Special Interest by Olvi L. Mangasarian, etc PDF

By Olvi L. Mangasarian, etc

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Read Online or Download Nonlinear Programming 2. Proceedings of the Special Interest Group on Mathematical Programming Symposium Conducted by the Computer Sciences Department at the University of Wisconsin–Madison, April 15–17, 1974 PDF

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Additional info for Nonlinear Programming 2. Proceedings of the Special Interest Group on Mathematical Programming Symposium Conducted by the Computer Sciences Department at the University of Wisconsin–Madison, April 15–17, 1974

Example text

Using a example k k 2 y and $'|y| . Hence, classical result (cf. for [1] page 200), and setting: 45 PIERRE HUARD φ κ (θ) = max ίφ κ (θ) |θ > 0} we have : k+1 k f( x ) - f(x) = max ίφ]ζ(θ) |θ€[0,1]}. r k k ,k,2 ±(Vf ( x ) . 2, with 0 < < 1, z 7* x , k the maximization of f on I x, zJ , and possibly a pivotal operation Setting C, = {x-,|x-, > 0}, we note that the intermediate steps (kh) are related to the projections of the different parts of the segment k k ïk Lxïk'xïk+\dk . ) l k=0 I k oo f k k+1 < f( x ) - f(x) k Σ |y| < + - k=0 « k+l_k Σ | x x | < + °o because k=0 «> k+1 k Σ ( x -x) is a convergent k=0 and hence sum.

54 A Q U A S I - N E W T O N M E T H O D FOR MINIMIZATION UNCONSTRAINED PROBLEMS by Klaus R i t t e r ABSTRACT A m e t h o d is d e s c r i b e d for the m i n i m i z a t i o n of a f u n c t i o n F ( x ) of n v a r i a b l e s . C o n v e r g e n c e to a s t a t i o n a r y point is shown w i t h o u t a s s u m p t i o n s on second o r d e r d e r i v a t i v e s . If the s e q u e n c e g e n e r a t e d by this m e t h o d has a c l u s t e r point in a n e i g h b o u r ­ hood of w h i c h F ( x ) is t w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e and has a p o s i t i v e d e f i n i t e H e s s i a n m a t r i x , then the c o n v e r g e n c e is s u p e r l i n e a r .

It follows from Lagrange's j J J J formula [11] that, for every choice of σ · > 0 , j do) |c;jd. l c n j ( g j - gj+l>l I |Sj jsj where ξ. e {x|x = x. - t o J, s J. , 0 4 t 4 1 } . From J J (10) it is clear that (9) need not always be satisfied. |a In [9] the inequality (9) is used to test whether d. is acceptable as the first column of J LV' + l* If (9) is not satisfied d. is rejected anc replaced by gj - v F ( X j *J ■ '°jsj 'nj 'nj -) l°jsj Thus an additional gradient evaluation is necessary whenever (9) is not satisfied.

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