For the Logistic model, why is the objective function unbounded below if two sets are linearly seperated? I am reading Approximate linear discrimination via logistic modeling in the Section 8.6.1 of B & V's Convex Optimization book. On Page 428,
$$
\operatorname{minimize} \  -l(a, b)   \tag{8.27}
$$
with variables $a$, $b$, where $l$ is the log-likelihood function
$$
\begin{aligned}
&l(a, b)=\sum_{i=1}^{N}\left(a^{T} x_{i}-b\right)-\sum_{i=1}^{N} \log \left(1+\exp \left(a^{T} x_{i}-b\right)\right)-\sum_{i=1}^{M} \log \left(1+\exp \left(a^{T} y_{i}-b\right)\right)
\end{aligned}
$$
It says that if two sets can be linearly seperated, i.e., if there exist $a$, $b$ with $a^T x_i > b$ and $a^T y_i < b$, then the optimization problem (8.27) is unbounded below. Why is it unbounded below for this case?

Think about the simplest case where there is a single predictor that is binary, i.e., you are using the binary logistic model to compute two proportions.  If P(Y=1 | X=0) = 1 and P(Y=1 | X=1) = 0 you have perfect separation.  The estimated proportions are 1.0 and 0.0 and there is no problem with the log likelihood function, which is zero.
