- The branch and bound method for solving an integer linear program or mixed integer linear program uses a search tree.
- At each node of the search tree, a linear program (LP) that relaxes some or all of the integrality restrictions (while adding some new restrictions based on earlier branching) is solved. (As an aside, this hold for mixed integer quadratic programs, with the change that the node relaxation will likely be a quadratic program rather than a linear program.)
- LPs have closed and convex feasible regions.
- The act of branching (separating a node into two child nodes) partitions the feasible region for the problem at the parent node into two smaller feasible regions.

Figure 1: Feasible region |

A general approach for partitioning a feasible region is to introduce one binary variable for each subregion/scenario, and constrain those variables to add up to 1 (so that exactly one scenario is selected). In our example, we add $z_i \in \{0,1\}$ for $i \in \{1,2,3\}$ along with the constraints\begin{eqnarray*} z_{1}+z_{2}+z_{3} & = & 1\\ x & \le & 3z_{1}+9(1-z_{1})\\ x & \le & 6z_{2}+9(1-z_{2})\\ x & \ge & 4z_{2}\\ x & \ge & 7z_{3}. \end{eqnarray*}Left to the reader as an exercise: confirm that if $z_1=1$ then $0\le x \le 3$, if $z_2=1$ then $4\le x\le 6$, and if $z_3=1$ then $7\le x\le 9$.

As a side note, before proceeding further, suppose that what we really want is to ensure that either $x\le 3$ or $x\ge 7$. (Don't ask why; just play along.) We do that the same way I just described, but then also constrain $z_2 = 0$. So the technique for partitioning a feasible region also works for omitting a chunk of a feasible region.

One limitation of this approach is that the subregions must be definable using

*linear*constraints, assuming that you want your problem to remain a MILP. Another (and this is really what motivated me to do this post) is that the partitioning really takes place

*during the branching process*. The algebra splits our feasible region into three disjoint chunks so long as $z_1,z_2,z_3$ are integer, but when we are solving node relaxations some or all of the $z$ variables will be relaxed to continuous.

Let's stick with our three-way split, while also fixing $z_2=0$, so that we force either $x\le 3$ or $x\ge 7$. The conceptual feasible region looks like Figure 2. Again, the actual feasible region for the MILP consists of two clusters of vertical line segments; the shaded polytopes are the relaxations of each scenario.

Figure 2: Two scenarios |

What about before any of the $z$ variables are fixed (for instance, at the root node)? Since $z_1$ and $z_3$ can take fractional values, the relaxation at the root node (shown in Figure 3) is the same as what it was before we introduced the $z$ variables.

Figure 3: Root node relaxation |

The key takeaway here is that the split we see in Figure 2 actually takes effect during branching, not at the root node (and not at nodes below the root but above the point where the solver branches on one of the $z$ variables). The approach of using binary variables to partition feasible regions is valid because the

*final solution*will obey the integrality restrictions (and thus will belong to one of the disjoint scenarios). Along the road to the final solution, though, we cannot count on the partitioning holding true (and, in the case of excluded regions, we cannot count on the unwanted portion actually be excluded yet).