Notice that IloCplex has methods by the same name for getting variable values and adding constraints to the model. However, using these methods in the middle of a callback will cause an error.

To make a

The method getValue(IloNumVar v) is significantly slower than getValues(IloNumVar[] vars), as previously discussed for IloCplex.

{gliffy:name=defaultCplex|align=left|size=L|version=5}

The algorithm terminates when the node stack is empty, and the best incumbent found is the new solution.  The first node pushed on, the LP relaxation of the original problem is often called the _root node_.


h1. Lazy Constraints in CPLEX

Lazy constraints, at a high level, are constraints that are only checked when a candidate for an integer solution has been identified.  They can be added to the model at the start though the {{IloCplex}} method {{addLazyConstraint(IloRange range)}}, or they can be added on the fly with a {{LazyConstraintCallback}} (documentation [here|http://pic.dhe.ibm.com/infocenter/cosinfoc/v12r5/index.jsp?topic=%2Filog.odms.cplex.help%2Frefjavacplex%2Fhtml%2Filog%2Fcplex%2FIloCplex.LazyConstraintCallback.html]).  The {{LazyConstraintCallback}} provides a user implemented routine to take a new potential incumbent solution, determine any of the lazy constraints were violated, and if so, to add at least one of them to the model.  Typically, you would want to use lazy constraints when you have a large number of constraints and you expect that few will be violated, as they allow you to have smaller LPs to solve when processing each node.  In the case of the {{LazyConstraintCallback}}, you gain an additional advantage in that you do not need to explicitly store all of the lazy constraints in memory, which is critical for problems like TSP that have an exponential number of constraints.  The downside is that you can waste a lot of time in branch and bound trying to generate integer solutions, only to find that they are actually infeasible.  Regardless of whether {{addLazyConstraint(IloRange range))}} or a {{LazyConstraintCallback}} was used, the model becomes

{gliffy:name=lazyCplex|align=left|size=L|version=2}

h1. Using LazyConstraintCallback

Qualitatively, the idea of the class {{LazyConstraintCallback}} is that you pass CPLEX the function meeting the following specification:
* *Input:* an integral solution to your IP that is guaranteed to satisfy all constraints except the Lazy Constraints not yet added
* *Output:* a (potentially empty) list of violated constraints that is guaranteed to be nonempty if at least one constraint was violated

However, in Java, we do not pass functions (methods), we pass objects satisfying some interface specifying methods.  This means we must create an object of type {{LazyConstraintCallback}}.  The class is {{abstract}}, (see [Java introduction|Do you know Java]), so we must extend the class and fill in the abstract methods, where we will specify how to check for violated constraints.  Since LazyConstraintCallback is an abstract class, it comes with some methods already implemented:

||Method Name||Return Type||Arguments||Description||
|getValue|double|IloNumVar var|Returns the solution value of var at the current node.|
|getValues|double[]|IloNumVar[] vars|Returns the solution values for vars at the current node.|
|add|IloRange|IloRange cut|Adds the constraint cut to the model.  Assumes cut is linear.|

To generate the constraints we will add, we use the same {{IloCplex}} instance that is solving the IP.
{warning:title=Warning}
Notice that {{IloCplex}} has methods *by the same name* for getting variable values and adding constraints to the model.  However, using these methods in the middle of a callback will cause an error.
{warning}
{info}
To make a {mathinline} \ge {mathinline} constraint in the middle of callback, be sure to call {{cplex.ge()}}, not {{cplex.addGe()}}, as discussed [here|Java and CPLEX#The Java Interface to CPLEX].
{info}
{warning:title=Warning}
The method {{getValue(IloNumVar v)}} is significantly slower than {{getValues(IloNumVar[] vars)}}, as [previously discussed for IloCplex|Java and Cplex#Performance Issues Reading Variable Values from CPLEX].
{warning}

It is convenient to make the new class an [inner class|http://docs.oracle.com/javase/tutorial/java/javaOO/nested.html] of the class where you are formulating your optimization problem, as then all of the fields which hold your problem data will be accessible.  In particular, you will need access to the {{IloCplex}} instance as well as whatever data structure you have storing your {{IloIntVar}} instances.

h1. TSP Integer "Solutions" Satisfying the Degree Constraints

To accomplish our task of identifying violated cutset constraints for integer solutions that are ensured to satisfy all of the degree constraints, we need to understand the combinatorial structure of these solutions to avoid checking all {mathinline}2^{|V|}{mathinline} constraints.  However, it is easy to see that in any (finite) graph where every node has degree two, the graph must be a collection of node disjoint cycles.  When the graph is a single cycle, none of the cutset constraints will be violated, and when it contains multiple cycles, we will have a violated constraint for each cycle, as there will be no edges across the cut separating the nodes in the cycle from the rest of the graph.  Thus given the {mathinline}|V|{mathinline} edges in a solution, we can identify all such cuts in {mathinline}O(|V|){mathinline} by partitioning the graph into connected components.  Note however that each of the cuts constraints will have potentially {mathinline}O(|E|){mathinline} nonzero coefficients.

h1. Implementing LazyConstraintCallback for the Cutset Constraints

First, we will extend {{LazyConstraintCallback}} with an innner class inside {{TspIpSolver}}.  At the bottom of {{TspIpSolver.java}} (but before the final '\}' ending the class), add the following code to extend {{LazyConstraintCallback}}:
{code}
	private class IntegerCutCallback extends LazyConstraintCallback{

		public IntegerCutCallback(){}

		@Override
		protected void main() throws IloException {
			//fill this in
		}			
	}
{code}

This solution is a little sloppy in that if there are only two connected components, the set of edges for their cut will be the same, but we will add the constraint twice. To get around this, you could cache the sets of edges in a new HashSet, to automatically filter out duplicates, then add all the constraints once all the edge sets have been determined. To think about: when there connected components, are all of the constraints needed, or do any of the constraints imply the final constraint?