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Unformatted text preview: 9.8 Cutting Planes 301 x 4 = 1 , x 5 (free) = : x 5 = 1 , x 4 (free) = :- 3 ( )- 3 ( 1 ) + ( ) + 2 ( 1 ) + 3 ( ) ≤ - 2 ,- 3 ( )- 3 ( 1 ) + ( ) + 2 ( ) + 3 ( 1 ) ≤ - 2 ,- 5 ( )- 3 ( 1 )- 2 ( )- 1 ( 1 ) + ( ) ≤ - 4 ,- 5 ( )- 3 ( 1 )- 2 ( )- 1 ( ) + ( 1 ) ≤ - 4 . For x 4 = 1, the first constraint is infeasible by 1 unit and the second constraint is feasible, giving 1 total unit of infeasibility. For x 5 = 1, the first constraint is infeasible by 2 units and the second by 2 units, giving 4 total units of infeasibility. Thus x 4 = 1 appears more favorable, and we would subdivide based upon that variable. In general, the variable giving the least total infeasibilities by this approach would be chosen next. Reviewing the example problem the reader will see that this approach has been used in our solution. 9.8 CUTTING PLANES The cutting-plane algorithm solves integer programs by modifying linear-programming solutions until the integer solution is obtained. It does not partition the feasible region into subdivisions, as in branch-and-bound approaches, but instead works with a single linear program, which it refines by adding new constraints. The new constraints successively reduce the feasible region until an integer optimal solution is found. In practice, the branch-and-bound procedures almost always outperform the cutting-plane algorithm. Nevertheless, the algorithm has been important to the evolution of integer programming. Historically, it was the first algorithm developed for integer programming that could be proved to converge in a finite number of steps. In addition, even though the algorithm generally is considered to be very inefficient, it has provided insights into integer programming that have led to other, more efficient, algorithms. Again, we shall discuss the method by considering the sample problem of the previous sections: z * = max 5 x 1 + 8 x 2 , subject to: x 1 + x 2 + s 1 = 6 , 5 x 1 + 9 x 2 + s 2 = 45 , x 1 , x 2 , s 1 , s 2 ≥ ....
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This note was uploaded on 01/26/2011 for the course IEOR 4004 taught by Professor Sethuraman during the Fall '10 term at Columbia.
- Fall '10