Abstracting a concept from a base - some data or a more concrete concept - means identifying those features that are constitutive of the concept, i.e. that are taken as crucial in subsuming or not an object under the concept, while at the same time ignoring (leaving unspecified) all those properties which the objects covered possess in addition but which are immaterial to the concept in question. For instance, we abstract the concept of a table from our experience with a set of tables. That concept does not comprise the color of the table(s). This is so despite the fact that all real tables have colors. The concept does not deny this, it just leaves it open. Abstraction is essentially a step in an inductive procedure, although it may be guided (deductively) by more general principles.
A construct of thinking is an idealization of some concept iff it changes or omits any of the features constituting that concept in order to simplify it. In an idealization, we assume a state of affairs that does not correspond to known reality. We do so in a methodological situation where our subject matter is so hopelessly complex that we are incapable of proposing a theory all of whose concepts are interrelated in such a way as to cover appropriately the interactions of the objects meant by them. In such a situation, we limit our epistemic interest by singling out a concept and disregarding part of its complexity. We might, e.g. construct a concept of a colorless table, i.e. a table that does not reflect light. That would be an idealization that is incompatible with our experience of tables, which teaches us that all tables have a color. (See elsewhere for the concept of language as an example.) Moreover, there is by definition no methodological procedure that would allow us to pass from the idealized concept to the basic concept (from a colorless table to a real table). If there were, the idealization would be unnecessary. An idealization cannot be arrived at inductively, it can only be deduced from axioms or (failing that) be stipulated.
A theory is an empirical theory, i.e. a theory of an object area existing independently of it, only if its concepts and theorems can be operationalized. The operationalization of a concept or theorem consists in specifying a set of procedures by which it is to be applied to some observable phenomena. That typically involves the specification of a test that some phenomenon must pass in order to be subsumed under the concept or, on the contrary, the specification of certain phenomena that would, if they occurred, falsify a certain theorem. Thus, operationalization of the concepts and theorems of a theory is an essential step in rendering it falsifiable and, thus, empirical. (See Operationalisierung for an example.)
Since an idealized theoretical construct is one that comprises features which contradict known reality, it is by definition neither falsifiable nor operationalizable. That means that one admits idealizations in the construction of a theory at the price of immunizing it against falsification, i.e. of depriving it of the status of an empirical theory. The question of whether such a theory should be pursued in a science is then, ultimately, a question of the epistemic interest of the people responsible for that scientific activity.