The University of Jordan Department of Mathematics. Branch and Cut

Transcription

1 The University of Jordan Department of Mathematics Heythem Marhoune Amina-Zahra Rezazgui Hanaa Kerim Sara Chabbi Branch and Cut Prepared and presented by : Sid Ahmed Benchiha Ibtissem Ben Kemache Lilia Benakkouche Sara Boutata Supervised and edited by : Dr. Baha Alzalg Integer and Combinatorial Optimization ( ) Spring 2018 Integer Optimization (University of Jordan) BRANCH AND CUT / 17

2 Sommaire 1 Introduction 2 Description of the Branch and Cut Method 3 4 Conclusion Integer Optimization (University of Jordan) BRANCH AND CUT / 17

3 Introduction Classification An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers which is very difficult to solve. However, there are three different types of algorithms : 1 Exact algorithms. 2 Approximation algorithms. 3 Heuristic algorithms. Integer Optimization (University of Jordan) BRANCH AND CUT / 17

4 Introduction Exact Algorithms Examples of exact algorithms designed and used to solve combinatorial 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 the Branch and Cut Method. Integer Optimization (University of Jordan) BRANCH AND CUT / 17

5 Description of the Branch and Cut Method Description The Branch and Cut Method is a combinatorial optimization algorithm for solving IPL. This method uses both the Branch and Bound Method and the Cutting Plane Method. In particular, we augment the formulation of subproblem with additional cuts, in order to improve the bounds obtained from the linear programming relaxations. We illustrate the method with an example. Integer Optimization (University of Jordan) BRANCH AND CUT / 17

6 An Example Consider the integer programming problem min 6x 1 5x 2 s.t. 3x (F ) 1 + x 2 11 x 1 + 2x 2 5 x 1, x 2 0 and both integers. Integer Optimization (University of Jordan) BRANCH AND CUT / 17

7 As illustrated in Figure 1. The feasible integer points are marked. The linear programming relaxation (or LP relaxation) is obtained by ignoring the integrality restrictions and is indicated by the polyhedron contained in the solid lines. Figure 1 : Two dimensional integer programming problem of our Ex. Integer Optimization (University of Jordan) BRANCH AND CUT / 17

8 The Branch and Cut Method ( first solves the linear programming relaxation, giving the point , ), with value There is now a choice : Should the LP relaxation be improved by adding a cutting plane, for example, x 1 + x 2 5, or should the problem be divided into two by splitting on a variable? If the algorithm splits on x 1, two new problems are obtained : (F 1 ) (F 2 ) min 6x 1 5x 2 s.t. 3x 1 + x 2 11 x 1 + 2x 2 5 x 1 3 x 1, x 2 0 x 1, x 2 Z. min 6x 1 5x 2 s.t. 3x 1 + x 2 11 x 1 + 2x 2 5 x 1 2 x 1, x 2 0 x 1, x 2 Z. Integer Optimization (University of Jordan) BRANCH AND CUT / 17

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11 The optimal solution to the original problem will be better than the solutions of these two subproblems. The solution of the linear programming relaxation of (F 1 ) is (3, 2), with the optimal value 28. This solution is integral, so it solves (F 1 ), and becomes the incumbent best known feasible solution. The LP relaxation of (F 2 ) has optimal solution (2, 3.5), with the optimal value This point is non-integral, so it does not solve (F 2 ), and hence it must be attacked further. Integer Optimization (University of Jordan) BRANCH AND CUT / 17

12 Assume the algorithm uses a cutting plane approach and adds the inequality 2x 1 + x 2 7 to (F 2 ). This is a valid inequality, for which it is satisfied by every integral point that is feasible in (F 2 ). Further, this inequality is violated by (2, 3.5), so it is a cutting plane. The resulting subproblem is : (F 3 ) min 6x 1 5x 2 s.t. 3x 1 + x 2 11 x 1 + 2x 2 5 x 1 2 2x 1 + x 2 7 x 1, x 2 0 and both integers. The LP relaxation of (F 3 ) has the optimal solution (1.8, 3.4) with the optimal value Integer Optimization (University of Jordan) BRANCH AND CUT / 17

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14 Notice that the optimal value for this modified relaxation is larger than the value of the incumbent solution. The value of the optimal integral solution for the second subproblem must be at least as large as the value of the relaxation. Therfore, the incumbent solution is batter than any feasible integral solution for (F 3 ), so it actually solves the original problem. Integer Optimization (University of Jordan) BRANCH AND CUT / 17

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17 Conclusion Conclusion There are some methods to solve the mixed-integer linear programming. The Gomory Cutting Plane Method is fast, but unreliable. Branch and Bound Method is reliable but slow. The Branch and Cut Method combines the advantages from these two methods and improve the defects. It has proven to be a very successful approach for solving a wide variety of integer programming problems. We can solve the MILP by taking some cutting planes before apply the whole system to the branch and bound. Branch and Cut Method is not only reliable, but also faster than branch and bound alone. Finally, we understand that using Branch and Cut Method is more efficient than using the Branch and Bound Method. Integer Optimization (University of Jordan) BRANCH AND CUT / 17