Consider the following optimization problem: \begin{equation} \max_{\mu \in \mathcal{M}} \int \log\left( \int e^{\alpha U(x,y)} d\mu(y) \right) d\nu(x) \end{equation} where $\mathcal{M}$ is the space of probability measures on the real line endowed with the Borel $\sigma$-algebra, and $\nu \in \mathcal{M}$ is given. $U$ is a $\mathcal{C}^\infty$ function, is strictly concave in $y$, and $y \mapsto U(x,y)$ is uniquely maximized at $y=x$.

**Here is the question:** Is it possible to characterize the solution of this optimization problem in terms of $\nu$ and $U$, and in particular when $\alpha \to \infty$? If so, how?

**Remarks:**

(a) Intuitively, $\mu$ should put most mass where $\nu$ does and where $U(x,x)$ is largest. However, I was unable to show a result that makes this statement precise.

(b) I was able to show that, if $U$ is analytic in both its arguments, the support of $\mu$ is either (i) discrete with no accumulation point other than infinity, either (ii) the support of $\mu$ is the same as the support of $\nu$. In particular, if $\nu$ has bounded support, $\mu$ will generically be supported on a finite set of points.

(c) One can see analogies with Kullback-Leibler divergence minimization/entropy maximization, but I am relatively new to these questions and I was unable to relate both problems in a precise way. One related although a priori different problem is the one here: Optimization over space of probability measures

(d) Feel free to impose additional structure on the problem. In particular, one can start with the case in which $\nu$ is absolutely continuous with respect to the the Lebesgue measure. However, note that the main difficulty with the problem at hand is that $\mu$ may well not be absolutely continuous, which occurs in many cases.

I am new to information theory and related divergence minimization problems, so maybe this question has an easy answer. In any case any input will be greatly appreciated! Thank you very much in advance!