Let $(X,P)$ be a probability measure space. Let $K$ be a convex compact subset of $\mathbb R^d$ and let $F:X \to 2^{K}$ be a set-valued map. Assume that $F$ is:
- closed (i.e $F(x)$ is closed for every $x \in X$).
- $P$-integrabily bounded, i.e there exists $h \in L^1(X \to \mathbb R_+,P)$ such that $$ \|z\| \le h(x),\text{ for every }x \in X,\,z \in F(x). $$
- "totally measurable".
Let $A \subseteq \mathbb R^d$ be the Aumann integral of $F$ over $X$, w.r.t $P$, i.e., $$ A:=\int_X F\,dP = \left\{\int_X f\,dP \,\Big |\, f \in \mathfrak F\right\}, $$ where $\mathfrak F$ be the collection of functions $f \in L^1(X \to \mathbb R^d,P)$ such that $f(x) \in F(x)$ for $P$-almost all $x \in X$.
From Theorem 3.11 of Bambucini (1999), we know that $A$ is a convex compact subset of $\mathbb R^d$.
Let $\phi:\mathbb R^d \to \mathbb R$ be a Lipschitz-smooth convex function.
Question. Is there a general framework for minimizing $\phi$ over $A$ (preferrably, with rates of convergence) ?
I'm particularly interested in the case where $F(x) = M(x)K$, for some matrix-valued function $M:X \to \mathbb R^{n \times d}$ such that $x \mapsto \|M(x)\|_{op}$ is $P$-integrable.
Note. As a starting point, one could consider the case where $\phi$ is a linear function, i.e. $\phi(z) \equiv z^\top w$ for some fixed $w \in \mathbb R^d$.