Integer Programming

Integer programmes are almost identical to linear programmes with one very important exception. Some of the decision variables in integer programmes may need to have only integer values. The variables are known as integer variables. Since most integer programmes contain a mix of real variables and integer variables they are often known as mixed integer programmes. While the change from linear programming is a minor one, the effect on the solution process is enormous. Integer programmes can be very difficult problems to solve and there is a lot of current research finding “good” ways to solve integer programmes.

Integer programming (the process of solving a (mixed) integer programme) was originally done using the branch-and-bound process. The branch part of the process eliminated non-integer values for integer variables in the following way:

• Initially, all variables are left as real variables. The problem is solved using linear programming;
• If one of the integer variables in the linear programming solution has a fractional value, e.g., $x_i = 4.5$, then the linear programme is split in two and the fractional region eliminated. This is done by branching on the variable value, e.g., adding the constraint $x_i \leq 4$ to form one linear programme and $x_i \geq 5$ to form the other.
• By finding the optimal solution in each of these new linear programmes and comparing we can find the optimal solution for the original problem.
• If either of the new linear programmes has a fractional value for an integer variable then a new branch is needed.

This branching process results in the formation of a branch-and-bound tree (we will discuss the bounding next). Each node in this tree represents a linear programme consisting of the original linear programme and the extra branches added. Eventually all the leaf nodes in the tree will contain solutions where all the integer variables have integer values and no further branching is needed. All these values can be compared and the best one is the solution to the original integer programme.
*Note* For MIPs of any reasonable size this tree could be huge, in fact it grows exponentially as the number of integer variables increases.

The bounding process allows sections of the branch-and-bound tree to be removed from consideration before all the leaf nodes have integer solutions. It relies on the following optimization principle:

Adding constraints to a mathematical programming will result in a deterioration of the optimal objective value.

This means that adding the branching constraints to the linear programmes at the branch-and-bound tree nodes will mean the resulting nodes will have an optimal objective function value that is equal to or worse than the optimal objective function value of the original linear programme. Thus the objective function values get worse the deeper into the tree you look. Since we are finding the integer solution in the branch-and-bound tree with the best objective value, we can use any integer solutions to bound the tree.

The current best integer solution is called the incumbent. After solving a linear programme at a leaf node of the branch-and-bound tree one of the following conditions holds:

• The linear programme is infeasible (no more branching is possible);
• The linear programme solution is an integer solution with a better objective value than the incumbent. The incumbent is replaced with this new solution;
• The linear programme solution has a worse objective than the incumbent. Any nodes created from this node will also have a worse objective than the incumbent. This node is bounded by the incumbent objective;
• The linear programme solution is fractional and has a better objective value than the incumbent. Further branching from this node is necessary to ensure an optimal solution is found.

Only the last condition requires more branching, all the other conditions result in the node becoming fathomed and no more branching is required from that node.

Example

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