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.
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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.
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.
union of the disjunctive sets
but it is also a cut!
The new “feasible” solution!
Hence, we have a linear description of the inequalities valid for Pi.How do we get disjunctive cuts in practice?
Generate a cutting plane by:
i) Requiring that inequality be valid, i.e. (a, b)Î Pi;
ii) Requiring that it cuts-off the current fractional point
zj = 0 if element j is not in the cover
Valid for all k
is valid for
Do not generate cuts at every node of the search tree
1. Only at the root node (cut-and-branch)
2. Only at the top k levels of the search tree
3. Only at the first k evaluated nodes (best-first search)
4. Every kth evaluated node (skip factor)
Delete inactive cuts
If the dual variable associated with a cut has been 0 for k consecutive iterations, then delete the cut and move it to the cut pool