Fenchel-duality and separably-infinite programs

In two recent papers Chabnes, Gbibie, and Kortanek studied a special class of infinite linear programs where only a finite number of variables appear in an infinite number of constraints and where only a finite number of constraints have an infinite number of variables. Termed separably-infinite programs, their duality was used to characterize a class of saddle value problems as a uniextremai principle. We show how this characterization can be derived and extended within Fenchel and Rockafellar duality, and that the values of the dual separably-infinite programs embrace the values of the Fenchel dual pair within their interval. The development demonstrates that the general finite dimensional Fenchel dual pair is equivalent to a dual pair of separably-infinite programs when certain cones of coefficients are closed