Fenchel duality theorem in multiobjective programming problems with set functions

On several results about convex set functions

“…The result is true if the algebraic relative interior of dom F e is nonempty and aff(dom F e ) is closed, that is ic (dom F e ) = ∅ in the terminology of [18]. Note that the above result is cited and used in [9,12].…”

“…We do not know if F e is proper even if dom F = A. Note that the above result is cited and used in [9,12].…”

“…In order to obtain the converse we need to have that (F (Ω n )) n has a limit in R whenever (Ω n ) n ⊂ dom F is such that χ Ω n → w * ξ ∈ L ∞ . Note that the above result is cited and used in [9,12].…”

“…Since f is w-lsc we obtain thatf is alsoŵ-lsc, and because f is w-usc at x 0 , f isŵ-usc at x 0 . Taking the operator T to be defined by (12), we observe that ker T = X 0 , and so T :…”

“…Our aim in this note is to point out those statements for which we have doubts and to provide counterexamples for some of them. Our main motivation in doing this is provided by the recent papers [1] (a survey one) and [12] where the majority of problematic statements are cited and used.…”