Explanation of branch-and-cut method in solving MIP.

Question

I'm reading about the branch and cut algorithm to solve a mixed-integer programming problem. The interface and the steps of the algorithm are as follow:

The extreme ray $e^*$, if exists, is from the representation of the polyhedron $P$ associated with the problem, as the Minkowsky sum of the convex hull of a set $V$ and a cone $E$, such as in the following theorem:

I have several questions:

1. My understanding is that the existence of the extreme ray would indicate that the solution set is infinite, yet how do we know the problem wouldn't have an optimal value in this case?

2. The branch and bound idea is to split the original MIP into smaller problems, adding them to the list of active problems $L$ and go through each one of them. What does this mean exactly? Does an active problem contain a subset of the inequalities that define the original MIP?

3. In the "else if" condition in $(2.a.\beta)$, if $\bar{x}_K$ is integer, does this already solve our problem? Could this be a typo?

4. What does the "check limit" line in $3.b.\beta$ do?

Answer 1

1. In the unbounded case, the algorithm returns an extreme ray in the direction of the objective, providing a certificate of unboundedness.
2. Each active problem (branch-and-bound node) contains all the original inequalities, in addition to the branching inequalities that led to that node.
3. In that case, you have found a better integer feasible solution (incumbent) than you had before, but you don't know whether it is optimal until you explore the rest of the tree. You might find an even better solution later.
4. This step avoids branching on variables whose ceiling or floor would exceed $N$ in absolute value. I think in most implementations $N=\infty$, effectively omitting this step.