$($Optional$)$ This problem is done in Jupyter notebook. Implement proximal

gradient method and subgradient method in Python. Compare their convergence

under proper settings on the following LASSO problem:

$mi{n}_{n{R}^n}f(x)$:$=\frac{1}{2}||Ax-b|{|}^2+||x|{|}_1.$

where $Ain{R}^{mn},bin{R}^m,$ and $>0$ are the following:

Take $m=64,n=256,={10}^{-3}.$ For any $1im$ and $1jn,$ the $($i$,$j$)-$th

entry of $A$ is given by

$A_{i,j}=\frac{1}{\sqrt[\phantom{2}]{m}}cos(\frac{2(i-1)(j-1){}^2}{m}).$

and $b=Az$ with $z_i=1$ for $i=12,35,38,129,136,150,170,207,232,243$ and $z_i=0$

otherwise.

Plot $f(x^k)$ vs$.k$ curves in log$-$log scale by the two algorithms in the same figure.

Properly label the axes, curves and title the figure.