GE330_lect17 - Lecture 17 Solving Integer Programming...

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Lecture 17 Solving Integer Programming Problems April 2, 2009
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Solving ILPs ILPs can be used to formulate a lot of practical problems, but they are in general very hard to solve (NP-hard). There are three classes of algorithms for ILP. Exact algorithms. - try to solve integer programs to provable optimality. - Branch and bound, Cutting plane, Branch and cut,... - Usually can not handle large scale problems. Heuristic algorithms: - Try to find satisfiable solutions. - Nearest neighbor, Tabu search, Genetic Algorithms,... - Can handle large scale problems but no performance guar- antee Approximation Algorithms: find solutions which are guaran- teed to be close to the optimal solution 2
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Branch and Bound Start from the LP relaxation of an ILP, namely, first drop the integrity constraints. Solve the LP relaxation to optimality, if the optimal solution is integer, then we are done. Otherwise, choose an integer variable whose value in the LP relaxation optimal solution is not integer, branch on this variable, and create several LP subproblems. Keep doing this, some of the branches can be fathomed. This is a systematic way of exploring all feasible solutions. Fathoming can help to reduce the search space. 3
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Branch and Bound: an Example Consider the following ILP max z = 5 x 1 + 4 x 2 s.t. x 1 + x 2 5 10 x 1 + 6 x 2 45 x 1 , x 2 nonegative integer To get the LP relaxation, we simple drop the requirement that x 1 and x 2 are integer. max z = 5 x 1 + 4 x 2 s.t. x 1 + x 2 5 10 x 1 + 6 x 2 45 x 1 , x 2 0 The optimal solution to the LP relaxation is x 1 = 3 . 75 , x 2 = 1 . 25 and z = 23 . 75. This solution is not integer, but it does provide an upper bound (if we have a minimization problem, then the LP relaxation provides a lower bound) of the optimal solution of the original ILP.
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