Composition of minmax problems [lcc-001Y]

Let Y be a real vector space, and let (X,Y,L), (Y^*,Z,L) be minmax problems. Then we can try to define a composite minmax problem L\circ _Y L'(x,z) = \sup _y \inf _{y'} L(x,y) + L(y',z) - y'(y)

For this composition to be associative relies on a strong duality property. We probably shouldn't want to treat this as well-defined unless it holds.

Note that by the convex duality stuff, the minmax problem Y^*,Y,\operatorname {ev} acts as an identity for this composition.

This may fit together into a double category type structure for minmax problems (maybe restricted to linear maps between the spaces).