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Unformatted text preview: 7.5 Cutting Planes Integer programming problems are typically hard to solve. Nonetheless, the state of the art today enables IP models to be solved sufficently well, so that it is a key element of the optimization toolkit. Consider the following integer program: maximize x + y subject to y ≤ 100 x , y + 100 x ≤ 101, x, y ≥ 0, integer. The main technique to solve integer programs is to use branchandbound, which starts by solving the linear programming relaxation. What does that linear programming relaxation look like in this case? Since there are only two variables, we can draw a graph of the feasible region, as shown below. (1.01,0) (1,1) (0,0) (1,0) (101/200,101/2) The optimal solution to this linear program is (101 / 200 , 101 / 2), which has objective function value equal to 51.005. This is not integer, so we have not solved the IP yet. We do know, however, that the LP optimal value is an upper bound on the IP optimal value. By considering the graph above, we also know quite a bit about the integer program. Feasible solutions are the points in shaded region that take 86 on integer values for both variables; they are the points on the socalled integer lattice. In this case, there are only 3 feasible integer solutions: (0,0), (1,0), and (1,1). Hence, the optimal value is 2, and the optimal solution is (1,1). Thus, we see that the upper bound found by solving the LP relaxation is horrible ! While this is a contrived example, it remains true that many natural IP formulations of realworld optimization problems also have such a large integrality gap . Of course, there are other ways to write down an integer program that has the same set of feasible integer solutions. Suppose that we want the linear programming relaxation to be as “good” as possible (whatever good means, for now). We know that the feasible region of any linear program has to be convex: that is, if two points are feasible, then the entire line segment connecting them must also be feasible. So, if the integer program feasible region contains the 3 points indicated above, then we know that all of the points in the region below must be feasible for any linear programming relaxation: this region is often called the convex hull of all feasible integer solutions. (1,1) (1,0) (0,0) Of course, this feasible region can also be described as those points ( x, y ) satisfying y ≤ x , x ≤ 1, and x, y ≥ 0. If we add the constraint x and y are integer to this description, then we have obtained an equivalent integer 87 programming formulation to our original one. However, this new formulation is much nicer than the original one, in that its linear programming relaxation has the property that each basic feasible solution is an integer solution. That is, its linear programming relaxation has the integrality property ....
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This note was uploaded on 04/06/2008 for the course ORIE 321 taught by Professor Shmoys/lewis during the Spring '07 term at Cornell.
 Spring '07
 SHMOYS/LEWIS

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