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

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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

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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.

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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

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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

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Lift-and-Project cuts A Mixed 0-1 Program { } min , , , , cx Ax b x x i p i = 0 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 = = 0 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

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The LP relaxation
The optimal “fractional” solution x

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One side of the disjunction 0 = i x x
1 = i x The other side of the disjunction x

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The union of the disjunctive sets x
The convex-hull of the union of the disjunctive sets x

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One facet of the convex-hull but it is also a cut!
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IP_IMA_2 - Branch-and-Cut Valid inequality an inequality...

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