IP_IMA_2 - Branch-and-Cut Valid inequality: an inequality...

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Unformatted text preview: Branch-and-Cut Valid inequality: an inequality satisfied by all feasible solutions Cut: a valid inequality that is not part of the current formulation Violated cut: a cut that is not satisfied by the solution to the current LP relaxation Branch-and-Cut Branch-and-cut is a generalization of branch-and-bound where, after solving the LP relaxation, and having not been successful in pruning the node on the basis of the LP solution, we try to find a violated cut. If one or more violated cuts are found, they are added to the formulation and the LP is solved again. If none are found, we branch. Branch-and-Cut Given a solution to the LP relaxation of a MIP that does not satisfy all the integrality constraints, the separation problem is to find a violated cut. Cut Classification General purpose Relaxation Problem specific Cut Classification General purpose: a fractional extreme point can always be separated Gomory cuts 0-1 disjunctive cuts Cut Classification Relaxation cuts: 0-1 knapsack set Continuous 0-1 knapsack set Node packing Cut Classification Problem specific: generally facets, derived from problem structure blossom inequalities for matching comb inequalities for TSP Lift-and-Project cuts A Mixed 0-1 Program { } min , , , , cx Ax b x x i p i = 0 1 1 its LP Relaxation min ~ ~ ~ ~ cx Ax b Ax b where includes all bounds with optimal solution x Lift-and-Project cuts Generate cutting planes for any mixed 0-1 program: Disjunction ~ ~ ~ ~ Ax b x Ax b x x i i i = = 1 {0,1} Description of P conv Ax b x Ax b x i i i = = = ~ ~ ~ ~ 0 1 Choose a set of inequalities valid for P i that cut off x The LP relaxation The optimal fractional solution x One side of the disjunction = i x x 1 = i x The other side of the disjunction x The union of the disjunctive sets x The convex-hull of the union of the disjunctive sets x One facet of the convex-hull but it is also a cut!...
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This note was uploaded on 03/01/2012 for the course ECON 102 taught by Professor Mehmet during the Spring '12 term at Abraham Baldwin Agricultural College.

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IP_IMA_2 - Branch-and-Cut Valid inequality: an inequality...

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