Branch and cut
Author: Lindsay Siegmundt, Peter Haddad, Chris Babbington, Jon Boisvert, Haris Shaikh (SysEn 6800 Fall 2020)
Steward: Wei-Han Chen, Fengqi You
Introduction - Lindsay
Methodology & Algorithm
Methodology - Haris
Algorithm - Peter
Branch and Cut for is a variation of the Branch and Bound algorithm. Branch and Cut incorporates Gomery cuts allowing the search space of the given problem. The standard Simplex Algorithm will be used to solve each Integer Linear Programming Problem (LP).
Below is an Algorithm to utilize the Branch and Cut algorithm with Gomery cuts and Partitioning:
Upper Bound = ∞ Lower Bound = -∞
Step 1. Initialize:
Set the first node as while setting the active nodes as .
Step 2. Terminate:
Step 3. Iterate through list L:
While L is not empty (i is the index of the list of L), then:
Step 3.1. Convert to a Relaxation:
Replace in by two constraints and
Solve for the Relaxed
If Z is infeasible: Return to step 3. else: Continue with solution Z.
Step 3.2. Cutting Planes:
If a cutting plane is found: then add to the linear Relaxation problem (as a constraint) and return to step 3.2 Else: Continue.
Step 4. Pruning and Fathoming:
If Z^l >= Z: return to step 3.
If Z^l <= Z AND X_i is an integral feasible: Z = Z^i Remove all Z^i from Set(L)
Step 5. Partition (reference EXTERNAL SOURCE)
Let be a partition of the constraint set of problem . Add problems to L, where is with feasible region restricted to and for j=1,...k is set to the value of for the parent problem l. Go to step 3.