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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 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 LP relaxation ILP problem LP problem Optimum Primal Simplex yes yes no UNSAT Feasible? Integral? Dual Simplex no Gomory’s derivation
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
Gomory’s derivation New slack variable
Cut from the Simplex tableau (I) • After primal Simplex method, consider the optimal tableau:
Cut from the Simplex tableau (II) New slack variable
Cut from the Simplex tableau (III) • Then apply dual Simplex method… nonnegative negative
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
Termination of the method (II) 1. each column in each tableau is lexicographically positive: i.e. y =
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 =
Termination of the method nonnegative
Termination of the method • Theorem Under the conditions above, the cutting plane method terminates. Proof:
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