Branch and bound method in case the non integer value is less than 1.

Question

If I need to find the integer valued solution to Linear programming problem using branch and bound method, but the non integer x value at the last iteration is less than 1, let's say 3/4. Then I have one constrain $x_1 \le 0$ and $x_1 \ge 1$. How do I proceed with the first case, when $x_1 \le 0$? I cannot add another constrain with auxiliary variable $x_1 + s_1 \le 0$, because all $x_i \ge 0$. Can I simply assume that $x_1 = 0$ and do the linear programming problem ignoring the $x_1$ variable?

Answer 1

If you started with nonnegative constraint and you branch on $x_1 \le 0$ and $x_1 \ge 1$, under that branch of $$x_1 \le 0$$ and $$x_1 \ge 0,$$ you can indeed conclude that $$x_1=0.$$

The value indeed has been fixed for that branch.