In these slides (5) the dual function for the norm minimization problem:
$$ \min_x \|x\| \quad \mbox{s.t.} \quad Ax =b $$
is defined as:
$$ g(v) = \inf_x (\|x\| - ν^\intercal Ax + b^\intercal ν) $$
what I don't understand is why the signs are reserved. The Lagrangian is according to the same author
$$ \|x\| + v^\intercal (Ax - b) $$
so the dual function should have been
$$ g(v) = \inf_x (\|x\| + ν^\intercal Ax - b^\intercal ν) $$
Is this correct?
$$g^{'}(v) = \inf_x (\|x\| + ν^\intercal Ax - b^\intercal ν)$$ is equivalent to $$g(v) = \inf_x (\|x\| - ν^\intercal Ax + b^\intercal ν)$$
except for the final $v^T$ that you will obtain which leads to $\min ||{x}||$. It means $x_{min}$ which minimizes $||x||$ conditioned on $ Ax =b$ will be exactly the same if you use $g^{'}(v)$ or $g(v)$