Given a H\"older family of functions $F$ and a finitely irreducible CGDMS $\Phi$ encoded by a symbolic representation $E_A^{\infty}$, one may associate to each coding $\omega\in E_A^{\infty}$ a Birkhoff average $\xi(\omega)$ called the $F$-exponent of $\omega$, should it exist. The ergodic optimization of these exponents by way of zero-temperature style limits is important to the characterization of Birkhoff spectra for CGDMSs. In this talk, we will introduce the objects at play in this optimization problem and the results which address this problem for a large collection of possible families $F$.