Converting linear BIP constraints into convex hull

Given a linear BIP $$\text{Minimize}\;\;\;c^Tx$$ $$\hspace{6.5mm}\text{Subject to}\;\;\;Ax\leq b$$ $$\hspace{38mm}x\in\{0,1\}^n$$

We can in theory convert the constraints to the convex hull constraint $Hx\leq d$ which allows us to perform a linear relaxation and find the solution to the linear BIP that way. Is there an efficient way to find this convex hull and put it in the aforementioned form?

The only algorithm I'm vaguely aware of involves solving an exponential number of auxiliary LPs to determine the extreme points of the set. The exponential number arises from noting that an $n$-cube has $2^{n}$ vertices.