46:25 The key point of the augumented lagrangian. "Making `u` large forces the |x-v| down"

This is the best explanation I've ever had in any optimization problem.

Thanks for helping and supporting students around the world

Thank you Charles, that is gold!

Thank you :)

I feel like the equation 9.19 in the book is more intuitive, you are just gradually upping the weight of lamda(multiplier) to do increase penalty on the constraint to make the problem better conditioned.  But I do also realize reorganizing it into the square also does not change the problem

Thank you so much sir , please if you  can give me the chapters because i really need to read the details

Sure, the details are in this book manuscript in Chapters 9 and 10.
https://engineering.purdue.edu/~bouman/publications/pdf/MBIP-book.pdf

Chapter 9 covers the basics of augmented Lagrangian, dual space methods, and ADMM, and
Chapter 10 covers Plug-and-Play and ADMM in more detail.

If you have suggestions for improvements, please let me know.