Given an optimization problem $X$, it is easy to construct a decision problem $Y$, such that there is a two-directional polynomial-time reduction between $X$ and $Y$. Therefore, we can define a class of optimization problems "$P^O$", which is the class of all optimization problems whose corresponding decision problem is in $P$; and "$NPH^O$", which is the class of all optimization problems whose corresponding decision problem is NP-hard.
But, problems in $NPH^O$ are not equal in terms of approximation. To be concrete, assuming $P\neq NP$, we can partition $NPH^O$ into two non-empty subsets: the problems that have an FPTAS, and the problems that do not.
My question is: is there any parallel partition of the class of NP-hard decision problems? For example, assuming $P\neq NP$, is there a natural class $C$ of decision problems, such that $C^O$ is exactly the class of optimization problems that have an FPTAS?
(by "natural" I mean that $C$ can be defined as a class of decision problems, without referring to optimization problems).