Convex optimization is defined here: https://cstheory.stackexchange.com/questions/22314/definition-of-convex-optimization-problem-by-stephen-boyd-and-lieven-vandenberghThe problem is NP-hard: https://mathoverflow.net/questions/92939/is-that-true-all-the-convex-optimization-problems-can-be-solved-in-polynomial-tiBut is anything known about the approximation complexity of the problem?  Does it have a PTAS (poly time approximation scheme)?  Or is there a proof that a PTAS is impossible unless P=NP?  Are there any known results on upper/lower bounds on its approximability?Any information will be much appreciated.
Approximability of convex programming (convex optimization)
approximation
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 $\epsilon > 0$. In particular, there is no PTAS.