# 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.:
$$y = \frac{1}{1 + \exp(x b)}$$.