Gomory Cuts-The How and the Why

Gomory Cuts-The How and the Why - Gomory's Cuts Page 1 of 7...

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Generating Gomory's Cuts for linear integer programming problems: the HOW and WHY A Gomory's Cut is a linear constraint with the property that it is strictly stronger than its Parent, but it does not exclude any feasible integer solution of the LP problem under consideration. It is used, in conjunction with the Simplex Method, to generate optimal solutions to linear integer programming problems (LIP). Formally the LP and LIP problems under consideration are as follows: We refer to LP as the linear programming relaxation of LIP. Needless to say, if the optimal solution to LP satisfies the integrality constraint of LIP then it must also be optimal with respect to LIP. In such a case we do not need any cuts, in fact we can completely relax and have a good cup of coffee. We are interested in Gomory's Cuts in cases where the optimal solution we have for LP does not satisfy the integrality constraint of ILP. That is, we are dealing here with a situation where the optimal solution to LP is such that in the final Simplex Tableau of the LP problem we have a row whose RHS value is not an integer. This row represents a linear equality constraint whose RHS value is not an integer. As an example, consider the minute ILP problem Observe that all the coefficient of the constraints - including the RHS values - are inegers. In the usual LP manner we rewrite this problem as follows: where s 1 and s 2 are the slack variables associated with the two functional constraints. Note that the integrality also applies to these variables. The optimal solution to the linear programming relaxation of this problem is x = (2.25,3.75). The final Simplex Tableau for this problem is as follows: LP: opt c t x subject to Ax = b, x >= 0 LIP: opt c t x subject to Ax = b, x >= 0 and integer z*: = max z = 5x 1 + 8x 2 s.t. x 1 + x 2 <= 6 5x 1 + 9x 2 <= 45 x 1 , x 2 >=0, and integer z*: = max z = 5x 1 + 8x 2 s.t. x 1 + x 2 + s 1 = 6 5x 1 + 9x 2 + s 2 = 45 x 1 , x 2 , s 1 , s 2 >=0, and integer BV x1 x2 s1 s2 RHS x1 1 0 2.25 -0.25 2.25 x2 0 1 -1.25 0.25 3.75 z 0 0 1.25 0.75 41.25 Page 1 of 7 Gomory's Cuts 3/1/2011 http://www.ms.unimelb.edu.au/~moshe/620-362/gomory/index.html
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The situation is described graphically in Figure 1. Figure 1 The second row in the final Simplex Tableau represents the following constraint: 0x 1 + x 2 - 1.25s 1 + 0.25s 2 = 3.75 This constraint generates the following Gomory's Cut: 0.75s 1 + 0.25s 2 >= 0.75 Don't panic if you do not understand where this constraint came from. We discuss the construction of such constraints downstairs. The point to note at the moment is that if we add this constraint to the LIP original problem, we obtain the following new ILP problem: This is illustrated in Figure 2. The broken blue line represents the hyperplane induced by this new constraint (in the x plane). z*: = max z = 5x
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Gomory Cuts-The How and the Why - Gomory's Cuts Page 1 of 7...

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