We define a class of optimization problems based on NP sets, where the optimum is taken over a polynomially bounded range (NPbOpt). We show that if such an optimization problem is based on one of the known NP complete sets, then it is hard for FP_{tt}^{NP}. Moreover, we will characterize FP_{tt}^{NP} as the class of functions that reduces to such optimization functions. We will call this property strong hardness.
The main question is whether these function classes are
complete for FP_{tt}^{NP}.
That is,
whether it is possible to compute
an optimal value for a given optimization problem in FP_{tt}^{NP}.
We show that these optimization problems are
complete for FP_{tt}^{NP}, if and only if
one can
compute membership proofs for NP sets in FP_{tt}^{NP}.
This indicates that the completeness question is a hard one.