Define
$$
F(\lambda,x):=g(x)+\lambda\int\limits\_{\partial\_e Y}f(y)d\mu\_x(y)
$$
where


* $Y$ is a convex space and $w^\*$-compact, moreover $Y$ forms a Bauer simplex (in particular $Y$ corresponds to some subset of some algebraic state space),
* its extreme boundary $\partial\_e Y$ is $w^\*$-compact,
* the points $x$ correspond to elements in $Y$,
* the measures $\mu\_x$ are probability measures (depending on $x$) and have support in $\partial\_e Y$,
* $g$ is a functional ($w^\*$-continuous and affine) on $Y$,
* $f$ is a continuous function on $\partial\_e Y$,
* $\lambda$ is a real parameter,
* $F(0,\cdot)$ is assumed to be a $w^\*$-continuous, affine functional on $Y$.


Then I consider
$$
R(\lambda):=\inf\_{x\in Y} F(\lambda,x),
$$ which can be shown to exist.


I am interested in computing the derivative of $R$ with respect to $\lambda$. For this, one clearly needs some smoothness of $\inf$.


**My question is**: under what precise conditions this derivative can be taken, and moreover which result will it produce? For example, given that $R$ is differentiable in $\lambda$ and that the set of minimizers corresponding to $R(0)$ (i.e. the points $x\in Y$ for which $R(0)=F(0,x)$) is nonempty, what can we say?


I thank you in advance!



Derivatives of infimum in variational problem
Let $t:=\lambda$, so that
$$R(t):=\inf\_{x\in Y}F(t,x).$$
Suppose that $\int\_{\partial\_e Y}f\,d\mu\_x$ is lower-semicontinuous in $x$ (with respect to the appropriate topology, which you appear to assume to be the $w^\*$-topology with respect to some unspecified duality). Then the set
$$Y\_t:=\{x\in Y\colon F(t,x)=R(t)\}$$
is a nonempty compact set, for each real $t$. Further, let
$$L\_t:=\Big\{\int\_{\partial\_e Y}f\,d\mu\_x\colon x\in Y\_t\Big\}.$$


Take now any real $t$.
If $L\_t$ is a singleton set $\{l\_t\}$, then $R$ is differentiable at $t$ and $R'(t)=l\_t$.
If $L\_t$ is not a singleton set, then $R$ is not differentiable at $t$.


These statements follow e.g. from Theorems 0.25 and 1.5 in the book by Levin "Convex analysis in spaces of measurable functions and its applications in mathematics and economics", Moscow: Nauka, 1985. (I have this book only in Russian.)


