$3.$ Formulate the following problems as LPs:

$($a$)$ minimize $||$Ax $$ b$||1$ subject to $||$x$||$ $1.$

$($b$)$ minimize $||$x$||1$ subject to $||$Ax $$ b$||$ $1.$

$($c$)$ minimize $||$Ax $$ b$||1+||$x$||$ $.$

In each problem, A in Rm$$n and b in Rm are given, and x in Rn is the optimization variable.

As a reminder, the $1-$norm and infinity norm of x in Rn are defined as

n

$||$x$||1=$ X $|$xi$|,$

i$=1$

$||$x$||$ $=$ max $|$xi$|$

i$=1,...,$n