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Gomory cutting plane method

Gomory cutting plane method. Jen-Chi Lin. Outline. Problem specification Algorithm Primal Simplex method Dual Simplex method Gomory ’ s derivation Cut from the Simplex tableau Termination of the method Reference. Problem specification. ILP Problem: LP-relaxation. Algorithm.

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Gomory cutting plane method

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  1. Gomory cutting plane method Jen-Chi Lin

  2. Outline • Problem specification • Algorithm • Primal Simplex method • Dual Simplex method • Gomory’s derivation • Cut from the Simplex tableau • Termination of the method • Reference

  3. Problem specification • ILP Problem: • LP-relaxation

  4. Algorithm LP relaxation ILP problem LP problem Optimum Primal Simplex yes yes no UNSAT Feasible? Integral? Dual Simplex no Gomory’s derivation

  5. Primal Simplex method

  6. Primal Simplex method

  7. Primal Simplex method

  8. Primal Simplex method

  9. Primal Simplex method

  10. Primal Simplex method

  11. Dual Simplex method (I) • Very similar to the Simplex method • Uses some of the negative components of the constant vector b to obtain a pivot entry all the components of the objective vector c are nonnegative some of the components of the constant vector b are negative

  12. Dual Simplex method (II)

  13. Dual Simplex method (III)

  14. Dual Simplex method (IV)

  15. Gomory’s derivation New slack variable

  16. Cut from the Simplex tableau (I) • After primal Simplex method, consider the optimal tableau:

  17. Cut from the Simplex tableau (II) New slack variable

  18. Cut from the Simplex tableau (III) • Then apply dual Simplex method… nonnegative negative

  19. Termination of the method (I) • Throughout the procedure of cut-adding and dual simplex method, following conditions aresatisfied: • each column in each tableau is lexicographically positive • the right-most column of the subsequent tableaux decreases lexicographically, at each dual pivot iteration • always take the highest possible row of the tableau as source row for the cut

  20. Termination of the method (II) 1. each column in each tableau is lexicographically positive: i.e. y =

  21. Termination of the method (III) 2. the right-most column of the subsequent tableaux decreases lexicographically, at each dual pivot iteration: i.e. y = z =

  22. Termination of the method nonnegative

  23. Termination of the method

  24. Termination of the method • Theorem Under the conditions above, the cutting plane method terminates. Proof:

  25. Termination of the method

  26. Termination of the method

  27. Termination of the method

  28. Termination of the method

  29. Reference • http://www.mcs.csueastbay.edu/~malek/Class/Dualsimplex.html • http://www.me.utexas.edu/~jensen/ORMM/methods/unit/linear/subunits/primal/primal_simplex.html • Theory of linear and integer programming, Alexander Schrijver,1986

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