I want to find dual problem of
$min_x||x-2||$ subject to $||x||_* \leq 1$
Here $x \in R^n$, $2$ denotes vector of $2s$, $||$ is arbitrary norm and $||_*$ is dual norm.
So I started with this:
$L(x,v) = ||x-2|| + v(||x||_*-1||)$
$g(v) = inf_x( ||x-2|| + v(||x||_*-1||))$
I don't know how to minimize this expression with respect to $x$
Reply to LinAlg's comment:
$min||y||$ subject to
$||x||_* \leq 1$, $y = x-2$
$L(y,v,k) = ||y|| + v(||x||_*-1)+k(y-x+2)$
$g(v,k) = inf_y(||y|| + v(||x||_*-1)+k(y-x+2))$
To minimize function above I need to minimize $||y|| + ky$ where $k \succeq 0$, but it seems to be unbounded below.