Minmax problems are not star-autonomous [lcc-001M]

Since the category of minmax problems is very similar to a Chu construction, we might hope that we could define a similar star-autonomous structure on minmax problems. Unfortunately, this does not work. We do have the duality, Dual minmax problem, but it doesn't extend to a star-autonomous structure.

Morally speaking, the tensor product of (X,Y,L), (X',Y',L') would be given by X \otimes X' in the forwards direction, and pairs of affine functions f: X \to Y', g: X' \to Y satisfying L(x,g(x')) = L'(x',f(x)) in the backwards direction, with either of these expressions giving the pairing. Since the first formula implies the pairing is convex in x (as it must be), but the latter implies it's concave in x, g must take values only those y so that L(-,y) is affine, and similarly for f. This can easily be an empty set, but in a star-autonomous category we always have a canonical costate L \otimes L^* \to I, which would be impossible in that case.