A Gradient Projection Algorithm for Relaxation Methods
A Gradient Projection Algorithm for Relaxation Methods
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In our eReader you can find the full English version of the book. Read A Gradient Projection Algorithm for Relaxation Methods Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book A Gradient Projection Algorithm for Relaxation Methods
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Combining, we have shown that w £ F^ . But X ■^ ^ V V 1 q'w= ) q. V. + ) q. (v. + rrr;— v. ) N = q-v + (t^-qi)v^ > q-v, since i e S implies q. 0. This is a contradiction, since q*v is maximum among v e F^ . Thus we must have v. = for i e S„. That is, v e W. 1 N Theorem; The output vector u solves Problem P.
Proof : According to the lemma, it suffices to show that u e F^, and q*u >_ q«v for all v e W n f^.
-10- Clearly, y calculated in step 3 of the algorithm satisfies y. = for i e Sj^. Further..., if i e D, i ft S^, then q^ >^ t^ according to the definition of S, and so y . = q . - t^ > . So y. > for all i e D. Finally Ey, = by direct computation. Thus y e W.
In fact, as can be easily seen, step 3 merely performs an orthogonal projection of q onto the subspace W. That is, for any v e W, -v - . A ^bxt:. '' (q - y) . V = 0. ;oqo- . TO. R. J;;.. :.
So q«v = y«v for all v e W.
The output vector u calculated in step 4 is simply a length normalization of y, and so u is in W also.
What to read after A Gradient Projection Algorithm for Relaxation Methods?
You can find similar books in the "Read Also" column, or choose other free books by J L Mohammed to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Combining, we have shown that w £ F^ . But X ■^ ^ V V 1 q'w= ) q. V. + ) q. (v. + rrr;— v. ) N = q-v + (t^-qi)v^ > q-v, since i e S implies q. 0. This is a contradiction, since q*v is maximum among v e F^ . Thus we must have v. = for i e S„. That is, v e W. 1 N Theorem; The output vector u solves Problem P.
Proof : According to the lemma, it suffices to show that u e F^, and q*u >_ q«v for all v e W n f^.
-10- Clearly, y calculated in step 3 of the algorithm satisfies y. = for i e Sj^. Further..., if i e D, i ft S^, then q^ >^ t^ according to the definition of S, and so y . = q . - t^ > . So y. > for all i e D. Finally Ey, = by direct computation. Thus y e W.
In fact, as can be easily seen, step 3 merely performs an orthogonal projection of q onto the subspace W. That is, for any v e W, -v - . A ^bxt:. '' (q - y) . V = 0. ;oqo- . TO. R. J;;.. :.
So q«v = y«v for all v e W.
The output vector u calculated in step 4 is simply a length normalization of y, and so u is in W also.
What to read after A Gradient Projection Algorithm for Relaxation Methods?
You can find similar books in the "Read Also" column, or choose other free books by J L Mohammed to read onlineMoreLess
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