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Gomory’s cutting plane algorithm for integer programming

Gomory’s cutting plane algorithm for integer programming. Prepared by Shin-ichi Tanigawa. Rounding does not give any useful result. We first solve the LP-relaxation. Optimize using primal simplex method. Optimize using primal simplex method. Optimize using primal simplex method.

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Gomory’s cutting plane algorithm for integer programming

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

  2. Rounding does not give any useful result

  3. We first solve the LP-relaxation

  4. Optimize using primal simplex method

  5. Optimize using primal simplex method

  6. Optimize using primal simplex method

  7. The optimal solution is fractional

  8. Generating an objective row cut (1)

  9. Generating an objective row cut (1) (weakening)

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

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

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

  13. The first cutting plane:

  14. A new slack variable is added:

  15. The new cut is added to the dictionary

  16. Re-optimize using dual simplex method

  17. Re-optimize using dual simplex method

  18. A new fractional solution has been found

  19. Generating a constraint row cut (1)

  20. Generating a constraint row cut (1) (weaken)

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

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

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

  24. The second cutting plane

  25. Add a new slack variable:

  26. The new cut is inserted into the optimum dictionary

  27. Re-optimize using dual simplex method

  28. The new optimum solution is integral

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