ILOG CPLEX C++ API 9.0 Reference Manual

No feasible integer solutions are ruled out by the cut, but some fractional solutions, for
example (0.0, 0.4, 1.0), can no longer be obtained in any LP or QP subproblems at the
nodes, possibly reducing the amount of searching needed.
The branch & cut method, then, consists of performing branches and applying cuts at
the nodes of the tree. Here is a more detailed outline of the steps involved.
First, the branch & cut tree is initialized to contain the root node as the only active
node. The root node of the tree represents the entire problem, ignoring all of the explicit
integrality requirements. Potential cuts are generated for the root node but, in the interest
of keeping the problem size reasonable, not all such cuts are applied to the model
immediately. An incumbent solution, that is, the best known solution that satisfies all the
integrality requirements, is established at this point for later use in the algorithm.
Ordinarily, no such solution is available at this stage, but the user can specify a starting
solution using the method setVectors. Alternatively, when solving a sequence of
modified problems, the user can set the parameter MIPStart to the value 1 (one) to
indicate that the solution of the previous problem should be used as a starting incumbent
for the present problem.
When processing a node, IloCplex starts by solving the continuous relaxation of its
subproblem. that is, the subproblem without integrality constraints. If the solution
violates any cuts, IloCplex may add some or all of them to the node problem and
resolves it. This procedure is iterated until no more violated cuts are detected (or
deemed worth adding at this time) by the algorithm. If at any point in the addition of
cuts the node becomes infeasible, the node is pruned (that is, it is removed from the
tree).
Otherwise, IloCplex checks whether the solution of the node-problem satisfies the
integrality constraints. If so, and if its objective value is better than that of the current
incumbent, the solution of the node-problem is used as the new incumbent. If not,
branching will occur, but first a heuristic method may be tried at this point to see if a
new incumbent can be inferred from the LP/QP solution at this node, and other methods
of analysis may be performed on this node. The branch, when it occurs, is performed on
a variable where the value of the present solution violates its integrality requirement.
This practice results in two new nodes being added to the tree for later processing.
Each node, after its relaxation is solved, possesses an optimal objective function value
Z. At any given point in the algorithm, there is a node whose Z value is better (less, in
the case of a minimization problem, or greater for a maximization problem) than all the
others. This Best Node value can be compared to the objective function value of the
incumbent solution. The resulting MIP Gap, expressed as a percentage of the incumbent
solution, serves as a measure of progress toward finding and proving optimality. When
active nodes no longer exist, then these two values will have converged toward each
other, and the MIP Gap will thus be zero, signifying that optimality of the incumbent has
been proven.
ILOG CPLEX C++ API 9.0 REFERENCE M ANUAL 9