Boyd and Vandenberghe say that a convex optimization problem is one of the form:
minimize $f_0(x)$ subject to $$f_i(x)\le 0, i=1,\ldots m$$ $$a_i^\top x=b_i, i=1,\ldots p$$
where $f_0, \ldots,f_m$ are convex functions and the equality constraint functions must be affine.
My question is, what stops me from converting the equality constraint functions into inequalities so that I am no longer restricted to affinity with those functions? (any equality $a=b$ can be expressed as $a\le b$ and $a \ge b$).