The ILOG CPLEX Mixed Integer Optimizer contains a wealth of features intended to aid in the solution of challenging MIP models. While default strategies are provided that solve the majority of models without user involvement, there exist difficult models that benefit from attention to performance tuning. This section discusses the ILOG CPLEX parameters that are the most likely to offer help on such models.

Because many of these parameter settings directly affect the branch & cut algorithm, here is a description of how that algorithm is implemented within ILOG CPLEX.

Branch & Cut

In the branch & cut algorithm, ILOG CPLEX solves a series of continuous subproblems. To manage those subproblems efficiently, ILOG CPLEX builds a tree in which each subproblem is a node. The root of the tree is the continuous relaxation of the original MIP problem.

If the solution to the relaxation still has one or more fractional-valued integer variables after ILOG CPLEX tries to add cuts, then ILOG CPLEX branches on a fractional variable to generate two new subproblems, each with more restrictive bounds on the branching variable. For example, with binary variables, one node will fix the variable at 0 (zero), the other, at 1 (one).

The subproblems may result in an all-integer solution, in an infeasible solution, or another fractional solution. If the solution is fractional, ILOG CPLEX repeats the process.

ILOG CPLEX cuts off nodes when the value of the objective function associated with the subproblem at that node is worse than the cutoff value.

You set the cutoff value by means of the CutUp parameter (for a minimization problem) or the CutLo parameter (for a maximization problem), to indicate to ILOG CPLEX that integer feasible solutions worse than this cutoff value should be discarded. The default value of the lower cutoff is -1e+75 ; the default value of the upper cutoff is 1e+75. The defaults, in effect, mean that no cutoff is to be supplied. You can supply any number that you find appropriate for your problem. It is never required that you supply a cutoff, and in fact for most applications is it not done.

ILOG CPLEX will use the value of the best integer solution found so far, as modified by the tolerance parameters ObjDif or RelObjDif (absolute and relative objective function difference, respectively), as the cutoff. Again, it is not typical that users set these parameters, but they are available if you find them useful. Use care in changing these tolerances: if either of them is nonzero, you may miss the optimal solution by as much as that amount. For example, in a model where the true minimum is 100 and the absolute cutoff is set to 5, if a feasible solution of say, 103 is found at some point, the cutoff will discard all nodes with a solution worse than 98, and thus the solution of 100 would be overlooked.

Periodically during the branch & cut algorithm, ILOG CPLEX may apply a heuristic process that attempts to compute an integer solution from available information, such as the solution to the relaxation at the current node. This activity does not replace the branching steps, but sometimes is able to inexpensively locate a new feasible solution sooner than by branching, and a solution found in this way is treated in the same way as any other feasible solution. At intervals in the tree, new cuts beyond those computed at the root node may also be added to the problem.

Once ILOG CPLEX finds an integer solution, it does the following:

During the branch & cut algorithm, ILOG CPLEX may choose to continue from the present node and dive deeper into the tree, or it may backtrack (that is, begin a new dive from elsewhere in the tree). The value of the backtrack parameter, BtTol, influences this decision, in terms of the relative degradation of the objective function caused by the branches taken so far in this dive. Setting BtTol to a value near 0.0 increases the likelihood that a backtrack will occur, while the default value near 1.0 makes it more likely that the present dive will continue to a resolution (fathoming either via a cutoff or an infeasible combination of branches or the discovery of a new incumbent integer feasible solution). See the reference manual ILOG CPLEX Parameters for more details about how this parameter influences the computation that determines the decision to backtrack.

When ILOG CPLEX backtracks, there usually remain large numbers of unexplored nodes from which to begin a new dive. The node selection parameter, NodeSel, determines this choice.

In instances where Best Estimate node selection (NodeSel = 2 or 3) is in effect, the BBInterval parameter determines the frequency at which backtracking is done by Best Bound anyway. The default value of 7 works well, but you can set it to 0 (zero) to make sure that Best Estimate is used every time backtracking occurs.

After a node has been selected, the variable selection parameter, VarSel, influences which variable is chosen for branching at that node.

After a variable has been selected for branching, the BrDir parameter influences the direction, up or down, of the branch on that variable to be explored first.