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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 polynomialtime 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 • Let’s 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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 Fall '08
 Ralphs
 Operations Research, Linear Programming, Optimization, integer linear program, cij xij, aj yj

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