MIT15_053S13_lec10

13 integer programs integer programs a linear program

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Unformatted text preview: . But first, we’ll review what we mean by integer programs. 13 Integer Programs Integer programs: a linear program plus the additional constraints that some or all of the variables must be integer valued. We also permit “xj ∈{0,1},” or equivalently, “xj is binary” This is a shortcut for writing the constraints: 0 ≤ xj ≤ 1 and xj integer. 14 Types of Integer Programs Mixed integer linear programs (MILPs or MIPs) Pure Integer Programs 0-1 Integer Programs xj ≥ 0 and integer for some or all j. xj ≥ 0 and integer for every j. xj ∈ {0,1} for every j. Note, pure integer programming instances that are unbounded can have an infinite number of solutions. But they have a finite number of solutions if the variables are bounded. 15 Goals of lectures on Integer Programming. Lectures 1 and 2 – Introduce integer programming – Techniques (or tricks) for formulating combinatorial optimization problems as IPs Lectures 3 and 4. – How integer programs are solved (and why they are hard to solve). • Rely on solving LPs fast • Branch and bound and cutting planes Lecture 5. Review and modeling practice 16 A 2-Variable Integer program maximize 3x + 4y subject to 5x + 8y ≤ 24 x, y ≥ 0 and integer What is the optimal solution? 17 The Feasible Region 5 Question: What is the optimal integer solution? 3 4 What is the optimal linear solution? 0 1 2 Can one use linear programming to solve the integer program? 0 1 2 3 4 5 A rounding technique that sometimes is useful, and sometimes not. 0 1 2 3 4 5 Solve LP (ignore integrality) get x=24/5, y=0 and z =14 2/5. Round, get x=5, y=0, infeasible! Truncate, get x=4, y=0, and z =12 Same solution value at x=0, y=3. Optimal is x=3, y=1, and z =13 0 1 2 3 4 5 0 0 1 1 2 2 3 3 4 4 5 5 Consider the feasible regions for the two integer programs on this slide. 0 1 2 3 4 max 3x + 4y s.t. 5 5x + 8y ≤ 24 x, y ≥ 0 and integer 0 1 2 3 max x+ 5 3x + 4y s.t. 4 y≤ 4 2x + 3y ≤ 9 x, y ≥ 0 and integer 20 Which of the following is false for the two integer programs on the previous slide? 1. The two models are the same in that they have the same feasible regions a...
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