Lecture20 - Introduction to Mathematical Programming IE406...

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Unformatted text preview: Introduction to Mathematical Programming IE406 Lecture 20 Dr. Ted Ralphs IE406 Lecture 20 1 Reading for This Lecture Bertsimas Sections 10.1, 11.4 IE406 Lecture 20 2 Integer Linear Programming An integer linear program (ILP) is the same as a linear program except that the variables can take on only integer values. If only some of the variables are constrained to take on integer values, then we call the program a mixed integer linear program (MILP). The general form of a MILP is min c x + d y s.t. Ax + By = b x, y x integer We have already seen a number of examples of integer programs. Product mix problem Cutting stock problem Integer knapsack problem Assignment problem Minimum spanning tree problem IE406 Lecture 20 3 How Hard is Integer Programming? Solving general integer programs can be much more difficult than solving linear programs. There in no known polynomial-time algorithm for solving general MILPs. Solving the associated linear programming relaxation results in a lower bound on the optimal solution to the MILP. In general, an optimal solution to the LP relaxation does not tell us anything about an optimal solution to the MILP. Rounding to a feasible integer solution may be difficult. The optimal solution to the LP relaxation can be arbitrarily far away from the optimal solution to the MILP. Rounding may result in a solution far from optimal. We can bound the difference between the optimal solution to the LP and the optimal solution to the MILP ( how ?). IE406 Lecture 20 4 Duality in Integer Programming Lets consider again an integer linear program min c x s.t. Ax = b x x integer As in linear programming, there is a duality theory for integer programs. We can dualize some of the constraints by allowing them to be violated and then penalizing their violation in the objective function. We relax some of the constraints by defining, for given Lagrange multipliers p , the Lagrangean relaxation Z ( p ) = min x X { c x + p ( A x- b ) } where X = { x Z n | A x = b, x } and A = [( A ) , ( A ) ] . IE406 Lecture 20 5 More Integer Programming Duality Z ( p ) is a lower bound on the optimal solution to the original ILP, so we consider the Lagrangean dual max Z ( p ) ....
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This note was uploaded on 08/06/2008 for the course IE 406 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture20 - Introduction to Mathematical Programming IE406...

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