Classification problem: custom minimization measure
I think what you are dealing with is a type I vs. type II error discussion. It seems that you want to avoid Type II error. 

One way to go is to consider $F_\beta$ for your performance metric. It is a modified version of $F_1$. As seen here : https://en.wikipedia.org/wiki/F1_score, $F_\beta$ can be formulated in terms of type I /type II error. You then need to select a $\beta$ based on how much avoiding type II error is important to you. This can be done by considerations on the cost of type II errors compared to the cost of type I error.