### Polyhedral Approaches to Mixed Integer Linear Programming [chapter]

Michele Conforti, Gérard Cornuéjols, Giacomo Zambelli
2009 50 Years of Integer Programming 1958-2008
This survey presents tools from polyhedral theory that are used in integer programming. It applies them to the study of valid inequalities for mixed integer linear sets, such as Gomory's mixed integer cuts. . Solving the LP relaxation, we get the optimal tableau: z + 1.16x 3 + 1.52x 4 = 28.16 The corresponding basic solution is x 3 = x 4 = 0, x 1 = 1.3, x 2 = 3.3 with objective value z = 28.16. Since the values of x 1 and x 2 are not integer, this is not a solution of (3). We can generate a cut
more » ... from the constraint x 2 + 0.8x 3 + 0.1x 4 = 3.3 using the following reasoning. Since x 2 is an integer variable, we have Since the left-hand-side is nonnegative, we must have k ≥ 0, which implies 0.8x 3 + 0.1x 4 ≥ 0.3 This is the famous Gomory fractional cut  . Note that it cuts off the above fractional LP solution x 3 = x 4 = 0.