Constraint to unconstraint optimization problem by subsitution Given the following convex optimization problem

$\min_{x,p} ||x|| - p$
subject to $p > 0$

Can I change the above to an unconstrained convex optimization problem by substituting $c = \log(p)$ and minimize

$\min_{x,c} ||x|| - \exp(c)$

If this is possible I am interested in literature about this topic, but I couldn't find much.
Edit: This is just a minimal example. The problem in the example is of course unbounded, but I think my full optimization problem adds more complexity than necessary to answer my question.

Yes, of course you can do this. If $\phi$ is a diffeomorphism, over the appropriate domains
$$\min_x\ f(x) = \min_y\ (f\circ \phi)(y).$$
I don't know a reference offhand unfortunately, but it is rather intuitive that this should work -- also you can see that $[J\phi]\nabla f = 0$ if and only if $\nabla f =0$ since $J\phi$ is invertible.
