Let $I = [0,1]$ denote the unit interval. Set $X := L_\infty(I)$. Let $f : X \times X \to \mathbb{R}$ be continuous and affine in each component. For example, if $w \in L_1(I)$ then $$f(u,v) = \int_I (c_1 u + c_2 v + c_{12} u v) w$$ is such as map (with $c_{*} \in \mathbb{R}$).
Suppose $A, B \subset X$ are two nonempty, closed, convex, bounded subsets. The example I have in mind is $A \times B = U_a \times U_b$ for some $0 < a, b < 1$, where $$U_t := \{ u \in X : 0 \leq u \leq 1 \text{ and } \int_I u \leq t \}.$$
Does the restriction $f|_{A \times B}$ assume its minimum (on $A \times B$)?
References to the literature are greatly appreciated.