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?


Here's an answer that takes a look at what actually happens to the fit. Consider the example of $x \in \mathbb{R}$, such that the data is linearly separable at $x=0$. We model the data using linear regression without an intercept, i.e.:
$y = \frac{1}{1 + \exp(x b)}$.
Now as you can see in the plot above, as b increases, the fit gets increasingly close to the true data (black points). Hence, the best fit would be for infinite b, and thus the problem is unbounded.
