Questions 30 and 32, exam 2018

Hi, I have some questions concerning the 2018 exam.

For question 30, I can easily show that \( \Delta x_t = M\Delta y_t \) if \( \Delta x_t \) and \( \Delta y_t \) are the Newton steps of the t-th iteration for g and h respectively, and not h and g. Is it an error, or is there something that I did not understand ?

For question 32, from the question 31 and the fact that \( ||Ax-b||^2 = || A ( x-y ) + Ay - b ||^2 \), I find the same equation as in the statement but with a factor 1/2 for \( ||Ax-b||^2 \) and \( ||Ay-b||^2 \). Thus it comes to my mind that this is actually \( f(x) = 1/2 ||Ax-b||^2 \) (note the factor 1/2) that is smooth with the largest eigenvalue of A^TA as smoothing parameter.
An other way to see this is to compute the Hessian of \( f(x) = ||Ax-b||^2\) , which is \( 2A^TA \), whose norm is bounded by 2 times the largest eigenvalue of \( A^TA \). Where do I make any mistake ?

Top comment

Q30: Yes, y should be for h and x for g. It's a bit confusing, but you can check this at initialization.

Q32: Yes, you're right. Thanks for reporting.

Page 1 of 1

Add comment

Post as Anonymous Dont send out notification