Approximability of convex programming (convex optimization)
You haven't defined convex optimization in general, and indeed it is not clear how one would encode a general convex optimization problem. However, to answer your question it suffices to consider the special case of copositive programming. It is known that Independent Set can be expressed as a copositive program, and this problem is NP-hard to approximate up to a factor of $n^{1-\epsilon}$ for every $\epsilon &gt; 0$. In particular, there is no PTAS.