ILOG CPLEX C++ API 9.0 Reference Manual

Concepts
Branch & Cut
IloCplex, like all CPLEX technologies, uses branch & cut search when solving
mixed integer programming (MIP) models. The branch & cut search procedure manages
a search tree consisting of nodes. Every node represents an LP or QP subproblem to be
processed; that is, to be solved, to be checked for integrality, and perhaps to be analyzed
further. Nodes are called active if they have not yet been processed. After a node has
been processed, it is no longer active. IloCplex processes active nodes in the tree
until either no more active nodes are available or some limit has been reached.
A branch is the creation of two new nodes from a parent node. Typically, a branch
occurs when the bounds on a single variable are modified, with the new bounds
remaining in effect for that new node and for any of its descendants. For example, if a
branch occurs on a binary variable, that is, one with a lower bound of 0 (zero) and an
upper bound of 1 (one), then the result will be two new nodes, one node with a modified
upper bound of 0 (the downward branch, in effect requiring this variable to take only the
value 0), and the other node with a modified lower bound of 1 (the upward branch,
placing the variable at 1). The two new nodes will thus have completely distinct solution
domains.
A cut is a constraint added to the model. The purpose of adding any cut is to limit the
size of the solution domain for the continuous LP or QP problems represented at the
nodes, while not eliminating legal integer solutions. The outcome is thus to reduce the
number of branches required to solve the MIP.
As an example of a cut, first consider the following constraint involving three binary
(0-1) variables:
20x + 25y + 30z