On the supremum in linear fractional programming with respect to any closed unbounded feasible set

We consider a constrained maximization problem with a linear fractional function f over any closed and unbounded set X. Starting the analysis from the behavior of unbounded sequences in X, we find necessary and sufficient conditions for the supremum to be finite. We point out that when f does not have maximum value, the supremum is not necessarily attained along a recession direction. It is known that this fundamental property holds in the classical fractional problem with a polyhedral feasible region and we give a new proof of this result by using our approach and the Representation Theorem.