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Integer Optimization (University of Jordan) BRANCH AND CUT 30-04-2018 1 / 17

The University of JordanDepartment of Mathematics

Branch and Cut

Supervised and edited by : Dr. Baha Alzalg

Prepared and presented by :

Heythem MarhouneAmina-Zahra RezazguiHanaa KerimSara Chabbi

Sid Ahmed BenchihaIbtissem Ben KemacheLilia BenakkoucheSara Boutata

Integer and Combinatorial Optimization (0301973)

Spring 2018

Sommaire

1 Introduction

2 Description of the Branch and Cut Method

3 Example of the Branch and Cut Paradigm

4 Conclusion


Introduction

Classification

An integer programming problem is a mathematical optimization orfeasibility program in which some or all of the variables are restrictedto be integers which is very difficult to solve. However, there arethree different types of algorithms :

1 Exact algorithms.

2 Approximation algorithms.

3 Heuristic algorithms.


Introduction

Exact Algorithms

Examples of exact algorithms designed and used to solvecombinatorial optimization problems :

1 Cutting Plane Method.

2 Branch and Bound Method.

3 Branch and Cut Method.

4 Dynamic Programming.

In this work, we present a method called theBranch and Cut Method.


Description of the Branch and Cut Method

Description

The Branch and Cut Method is a combinatorial optimizationalgorithm for solving IPL. This method uses both the Branch andBound Method and the Cutting Plane Method.In particular, we augment the formulation of subproblem withadditional cuts, in order to improve the bounds obtained from thelinear programming relaxations. We illustrate the method with anexample.


Example of the Branch and Cut Paradigm

An Example

Consider the integer programming problem

(F )

min −6x1 − 5x2s.t. 3x1 + x2 ≤ 11

−x1 + 2x2 ≤ 5x1, x2 ≥ 0 and both integers.



As illustrated in Figure 1. The feasible integer points are marked.The linear programming relaxation (or LP relaxation) is obtained byignoring the integrality restrictions and is indicated by the polyhedroncontained in the solid lines.

Figure 1 : Two dimensional integer programming problem of our Ex.



The Branch and Cut Method first solves the linear programming

relaxation, giving the point

(2 +

3

7, 3 +

5

7

), with value - 33− 1

7.

There is now a choice : Should the LP relaxation be improved byadding a cutting plane, for example, x1 + x2 ≤ 5, or should theproblem be divided into two by splitting on a variable ?If the algorithm splits on x1, two new problems are obtained :

(F1) (F2)

min −6x1 − 5x2s.t. 3x1 + x2 ≤ 11

−x1 + 2x2 ≤ 5x1 ≥ 3

x1, x2 ≥ 0x1, x2 ∈ Z.

min −6x1 − 5x2s.t. 3x1 + x2 ≤ 11

−x1 + 2x2 ≤ 5x1 ≤ 2

x1, x2 ≥ 0x1, x2 ∈ Z.



The optimal solution to the original problem will be better than thesolutions of these two subproblems. The solution of the linearprogramming relaxation of (F1) is (3, 2), with the optimal value −28.This solution is integral, so it solves (F1), and becomes theincumbent best known feasible solution. The LP relaxation of (F2 )has optimal solution (2, 3.5), with the optimal value −29.5. Thispoint is non-integral, so it does not solve (F2), and hence it must beattacked further.



Assume the algorithm uses a cutting plane approach and adds theinequality 2x1 + x2 ≤ 7 to (F2). This is a valid inequality, for which itis satisfied by every integral point that is feasible in (F2). Further,this inequality is violated by (2, 3.5), so it is a cutting plane. Theresulting subproblem is :

(F3)

min −6x1 − 5x2s.t. 3x1 + x2 ≤ 11

−x1 + 2x2 ≤ 5x1 ≤ 2

2x1 + x2 ≤ 7x1, x2 ≥ 0 and both integers.

The LP relaxation of (F3) has the optimal solution (1.8, 3.4) with theoptimal value −27.8.



Notice that the optimal value for this modified relaxation is largerthan the value of the incumbent solution. The value of the optimalintegral solution for the second subproblem must be at least as largeas the value of the relaxation.Therfore, the incumbent solution is batter than any feasible integralsolution for (F3), so it actually solves the original problem.


Conclusion

Conclusion

There are some methods to solve the mixed-integer linearprogramming. The Gomory Cutting Plane Method is fast, butunreliable. Branch and Bound Method is reliable but slow. TheBranch and Cut Method combines the advantages from these twomethods and improve the defects. It has proven to be a verysuccessful approach for solving a wide variety of integer programmingproblems. We can solve the MILP by taking some cutting planesbefore apply the whole system to the branch and bound. Branch andCut Method is not only reliable, but also faster than branch andbound alone. Finally, we understand that using Branch and CutMethod is more efficient than using the Branch and Bound Method.


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