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Consider f e C} (R") and the following function 1 Hy (æ, z) = f(z) + Vf(z)"(x – z) + ||x – z||3, 27 where x, z E R" and 0 < y < 1/L. (a) Show that the following two statements are true Hy (x, z) > f(æ) for every æ, z E R" Hy (x, x) = f(x) for every æ E R".

Consider f e C} (R") and the following function 1 Hy (æ, z) = f(z) + Vf(z)"(x – z) + ||x – z||3, 27 where x, z E R" and 0 < y < 1/L. (a) Show that the following two statements are true Hy (x, z) > f(æ) for every æ, z E R" Hy (x, x) = f(x) for every æ E R".

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider f E C}(R") and the following function 1 Hy (x, z) = f(z) +Vf(z)"(x – z) + ||x – z|3, 27 - where x, z E R" and 0 < y < 1/L. (a) Show that the following two statements are true Hy (x, z) > f(æ) for every x, z E R" Hy(x, x) = f(æ) for every a e R". (b) Show that we can re-write the function My as 1 Hy (x, z) : l|x – (z – yVf(2)|3 + f(2) – „||Vf(2)|3. 27 (c) Show that the GM with a fixed step size y is equivalent to the following iterative algorithm t-1 arg min u, (x, x-'), t=1,2,3,... xĒR" (d) Using the arguments above, show that the GM iterations with 0 < y< 1/L statisfy f (x') < f(x'-1), t=1,2,3, ....
Transcribed Image Text:Consider f E C}(R") and the following function 1 Hy (x, z) = f(z) +Vf(z)"(x – z) + ||x – z|3, 27 - where x, z E R" and 0 < y < 1/L. (a) Show that the following two statements are true Hy (x, z) > f(æ) for every x, z E R" Hy(x, x) = f(æ) for every a e R". (b) Show that we can re-write the function My as 1 Hy (x, z) : l|x – (z – yVf(2)|3 + f(2) – „||Vf(2)|3. 27 (c) Show that the GM with a fixed step size y is equivalent to the following iterative algorithm t-1 arg min u, (x, x-'), t=1,2,3,... xĒR" (d) Using the arguments above, show that the GM iterations with 0 < y< 1/L statisfy f (x') < f(x'-1), t=1,2,3, ....
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