Definition Minmax problem [lcc-001C]

April 22, 2024
Eigil Fjeldgren Rischel

A minmax problem is a triple $(X,Y,L),$ where $X,Y$ are convex spaces (that is, algebras for the discrete distribution monad---more topological assumptions may be necessary here), and $L: X \times Y \to \mathbb {R}$ is a function which is

Pointwise convex in $X$ ---for each $y$ , given $x_1,x_2 \in X, \theta \in [0,1]$ , $L(\theta x_1 + (1-\theta )x-2,y) \leq \theta L(x_1,y) + (1-\theta )L(x_2,y)$
Pointwise concave in $Y$ ---for each $x$ , given $y_1,y_2 \in Y, \theta \in [0,1]$ , $L(x,\theta y_1 + (1-\theta )y_2) \geq \theta L(x,y_1) + (1-\theta )L(x,y_2)$

A morphism of minmax problems $(X,Y,L) \to (X',Y',L')$ is a pair of functions $\phi ^+: X \to X'$ and $\phi ^-: Y' \to Y$ so that $L(x, \phi ^-(y')) \geq L'(\phi (x),y')$