NP-hardness of an optimization problem with real value
No, you can not say that your problem is NP-complete.

The reason is that it is not in NP: there is no Turing machine at all (let alone a poly-time (non-deterministic) one) that can solve it because TMs can not handle real numbers.

The reduction may still go through and show that the problem is NP-hard; a subset of the reals should certainly suffice to model the naturals in Vertex Cover or whatever NP-complete problem you use. So it is, in principle, possible to define a computable, poly-time reduction function that does the job. It will output naturals (or integers, rationals, ...), of course, but as long as you are willing to accept these as real numbers you are fine.

This reduction is likely to carry over to restricted versions of your problem (e.g. finite precision) which can be classified in the framework of computabilty/complexity theory.