I'm self-studying on cutting plane methods, and I'm reviewing the following problem from Bertsimas' book (see below). I know what cutting plane methods do, and how they eliminate infeasible solutions from the relaxed problem. But, I can think of several examples where a solution to the relaxed problem will not be eliminated by the cut proposed in this example.
Here's the problem description, and then I'll explain a few examples where I believe the proposed integer cut will not solutions to the LP relaxation (equation 11.2 below):
If $x^* = [0.5,0]$, for example, then I understand how the proposed integer cut $\sum_{j \in N} x_j \ge 1$ will eliminate such a solution. (This also applies for any other basic feasible solution that has 2 components whose first fractional component is less than $1$, and the other is 0). However, the integer cut will not eliminate the following solution: $x^*=[1.5,0]$, or for a problem of a different dimension, $x^*=[0.5,0.7,0]$, yet this is also has "at least one fractional basic variable."
How does such a cut eliminate these solutions?
Note that the index of summation in the cut runs only over the nonbasic variables. So if $x^* = [1.5, 0],$ then $x^*_1$ is basic, $x^*_2$ is nonbasic, $N=\lbrace 2 \rbrace$ and the proposed cut is $x_2 \ge 1,$ which does in fact cut off $x^*.$