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Gomory’s cutting plane algorithm for integer programming Prepared by Shin-ichi Tanigawa

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Rounding does not give any useful result

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We first solve the LP-relaxation

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Optimize using primal simplex method

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The optimal solution is fractional

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(1) Generating an objective row cut

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(weakening) (1) Generating an objective row cut

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(weakening) (1) (2) (for integers) Generating an objective row cut

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(weakening) (1) (2) (for integers) (2) - (1) Generating an objective row cut Cutting plane is violated by current optimum solution

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(weakening) (1) (2) (for integers) (2) - (1) (substitute for slacks) Generating an objective row cut

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The first cutting plane:

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A new slack variable is added:

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The new cut is added to the dictionary

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Re-optimize using dual simplex method

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A new fractional solution has been found

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(1) Generating a constraint row cut

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(1) (weaken) Generating a constraint row cut

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(1) (2) Generating a constraint row cut (valid for integers)

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(1) (2) (2) – (1) Generating a constraint row cut

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(1) (2) (2) – (1) Generating a constraint row cut

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The second cutting plane

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Add a new slack variable:

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The new cut is inserted into the optimum dictionary

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Re-optimize using dual simplex method

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The new optimum solution is integral

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