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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. Branch-and-Cut.
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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
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
with optimal solution Lift-and-Project cuts • A Mixed 0-1 Program • its LP Relaxation
Lift-and-Project cuts • Generate cutting planes for any mixed 0-1 program: • Disjunction • Descriptionof • Choose a set of inequalities valid for Pi thatcut off
One side of the disjunction
The other side of the disjunction
The convex-hull of the union of the disjunctive sets
One facet of the convex-hull but it is also a cut!
x The new “feasible” solution!
Hence, we have a linear description of the inequalities valid for Pi. How do we get disjunctive cuts in practice? • A cut a x >b is valid for Pi if and only if (a, b) satisfies
Approach 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
Relaxation Cuts • A valid inequality for a relaxation of a problem is also a valid inequality for the problem itself • Idea: Derive valid inequalities for common relaxations
Cover Inequalities • 0-1 integer set • Cover C • Minimal cover C • Cover inequality
Separation • Given a point find a cover C such that
Separation • Let zj = 1 if element j is in the cover zj = 0 if element j is not in the cover • If v < 1, then we identified a violated cover
Lifting • Find valid inequality • Case 1: xk = 0 Valid for all k • Case 2: xk = 1
Lifting • Define
Lifting • Let k N\C. Find k such that is valid for
Lifting • verified if • where
Lifting • Proposition: Let k N\C, then • is valid for
Sequential Lifting • We can repeat the same procedure to lift the other variables in N\C • Different lifting sequence can lead to a different lifted cover inequality!
Continuous 0-1 Knapsack • Mixed integer set • We can apply similar ideas and results concerning lifting to generate valid inequalities for MIPs
Node Packing Relaxation • Problem • Implications
Node Packing Relaxation • Conflict graph • Clique inequality
Cut generation • Clique inequalities • Odd-cycle inequalities
Cut Management • Cut generation takes time (even if we are not successful) • Cuts increase the size of the formulation (and thus increases the time it takes to solve the LP relaxation) • Only useful if it leads to reduced overall solution times !
Cut Management • When and how many violated cuts to add the current formulation ? • When and which cuts to delete from the current formulation ?
Cut Management • Do not generate cuts at every node of the search tree • Limit the rounds of cut generation per node • Limit the number of cuts generated per round of cut generation • Delete inactive cuts
Cut Management 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)
Cut Management 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
Cut Management Cut generator Active formulation Cut pool
Branch-and-Price • Branch-and-price is a generalization of LP based branch-and-bound specifically designed to handle integer programs that contain a huge number of variables
Branch-and-Price • Columns are left out of the LP relaxation because there are too many to handle efficiently and most of them will have their associated variable equal to zero in an optimal solution anyway • To check the optimality of an LP solution, a pricing problem is solved to try to identify columns with profitable reduced cost
Branch-and-Price • If profitable reduced cost columns are found, they are added and the LP relaxation is resolved • If no profitable columns are found, the LP solution is optimal • Branching occurs when the optimal LP solution does not satisfy the integrality conditions
Branch-and-Price • Branch-and-price applies column generation at every node of the branch-and-bound tree
Why use formulations with a huge number of variables ? • Compact formulation may have a weak LP relaxation • Compact formulation may have a symmetric structure • Provides a decomposition in master and pricing problem • Only choice
Complications • Conventional branching on variables may not be effective because fixing variables destroys the structure of the pricing problem • Column generation often converges slowly and solving LPs to optimality may be computationally prohibitive
