In this talk, we will talk about some optimization problems related to the Riemann zeta function and $L$-functions. In particular, we will talk about the distribution of the low-lying zeros of families of $L$-functions. We will see how we can use the one-level density theorems in the literature to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line. This is based on joint work with E. Carneiro and M. B. Milinovich.