Identifying approximate proper efficiency in an infinite dimensional space

The main idea of this article is to characterize approximate proper efficiency that is  a widely used optimality concept in multicriteria optimization problems that prevents  solutions having unbounded trade-offs. We analyze a modification of approximate proper  efficiency for problems with infinitely many objective functions. We obtain some necessary  and sufficient optimality conditions for this modification of approximate proper efficiency.  This modified version of approximation guarantees the general characterizations of approximate properly efficient points as solutions to weighted sum problems and modified  weighted Tchebycheff norm problems, even if there is an infinite number of criteria. The  provided proofs concerning the modified definition show that if the number of the objective  functions is infinite, then these results become invalid under the primary definition of  approximate proper efficiency.