Let $T:X\to X$ be a continuous map on a compact metrizable space, let $f:X\to \mathbb{R}$ be continuous, and let $W\subset C(X)$ be a closed subspace of continuous functions from $X$ to $\mathbb{R}$. We consider the set $M_W(X,T)$ of all $T$-invariant Borel probability measures $\mu$ such that $\int gd\mu =0$ for all $g$ in $W$. Then we consider optimization problems of the form $max\int fd\mu +\tau h(\mu ),$ where $\mu$ ranges over $M_W(X,T)$, $h(\mu )$ denotes the entropy of $\mu$ with respect to $T$, and $\tau$ is either $0$ or $1$. Our main results concern the basic properties of such optimization problems, including feasibility, geometry of the solution set, uniqueness of solutions, and realizability. This talk is based on ongoing joint work with Shengwen Guo (UNC Charlotte).