Comment on a note on duality gaps in linear programming over convex sets

In Ref. 1, Soyster has given a rather complicated proof of the absence of a duality gap, under a certain interiority condition, for a variant of a pair of optimization problems introduced by Ben-Israel, Charnes, and Kortanek (Ref. 2). A proof can be given directly (and under weaker conditions) by a simple application of a Lagrange multiplier theorem on convex programming in abstract spaces (Ref. 3).