  We consider the following frustrated optimization problem: given a prior
probability distribution $q$, find the distribution $p$ minimizing the relative
entropy with respect to $q$ such that $\textrm{mean}(p)$ is fixed and large. We
show that solutions to this problem are asymptotically Gaussian. As an
application we derive an $H$-type theorem for evolutionary dynamics: the
entropy of the (standardized) distribution of fitness of a population evolving
under natural selection is eventually increasing.
