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

The Branch and Cut is a methodology that is used to optimize linear problems that are integer based. This concept is comprised of two known optimization methodologies - branch and bound and cutting planes. Utilizing these tools allows for the Branch and Cut to be successful by increasing the relaxation and decreasing the lower bound. The ultimate goal of this technique is to minimize the amount of nodes.

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:

Step 0:

Upper Bound = ∞
Lower Bound = -∞

Step 1. Initialize:

Set the first node as $LP_{0}$ while setting the active nodes set as $L$ . The set can be accessed via $LP_{n}$

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:

Solve 3.2.

Solve for the Relaxed

Step 3.3.

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 $D_{j=1}^{lj=k}$ be a partition of the constraint set $D$ of problem $LP_{l}$ . Add problems $D_{j=1}^{lj=k}$ to L, where $LP_{j}^{l}$ is $LP_{l}$ with feasible region restricted to $D_{j}^{l}$ and $Z_{lj}$ for j=1,...k is set to the value of $Z^{l}$ for the parent problem l. Go to step 3.