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# hw3 - IE 495 Stochastic Programming Problem Sets#5#7 Due...

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IE 495 – Stochastic Programming Problem Sets #5—#7 Due Date: April 28, 2003 Do the following problems. If you work alone, you will receive a 10% bonus on your score. These are the final homework sets for the semester. Problems 1–2 will be Problem Set #5, Problems 3–4 will be Problem Set #6, and Problems 5–6 will be Problem Set #7. You are allowed to examine outside sources, but you must cite any references that you use. Please don’t discuss the problems with other members of the class (other than your partner, if you are working with one). 1 Feasibility Cuts The second stage constraints of a two-stage problem look as follows: 1 3 - 1 0 2 - 1 2 1 y = - 6 - 4 ω + 5 - 1 0 0 2 4 x y 0 . Here ω is a random variable with support Ω = [0 , 1]. 1.1 Problem Write down the linear programs (both primal and dual formulation(s)) needed to check whether or not there is a feasible second stage solution for a given x . 1.2 Problem Describe how these formulations allow you to obtain an inequality that cuts of x , if there is no feasible second stage solution y for that x . 1.3 Problem Let ˆ x = (1 , 1 , 1) T . Find the inequality explicitly for this first stage solution ˆ x .

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IE495 Problem Sets #5—#7 2 Bounds Consider our favorite random linear program. minimize Q ( x 1 , x 2 ) = x 1 + x 2 + 5 Z 4 ω 1 =1 Z 1 ω 2 =1 / 3 y 1 ( ω 1 , ω 2 ) + y 2 ( ω 1 , ω 2 ) 1 2 subject to ω 1 x 1 + x 2 + y 1 ( ω 1 , ω 2 ) 7 ω 1 , ω 2 Ω ω 2 x 1 + x 2 + y 2 ( ω 1 , ω 2 ) 4 ω 1 , ω 2 Ω x 1 0 x 2 0 y 1 ( ω 1 , ω 2 ) 0 ω 1 , ω 2 Ω y 2 ( ω 1 , ω 2 ) 0 ω 1 , ω 2 Ω Ω = { ω 1 × ω 2 } ω 1 ∼ U [1 , 4] ω 2 ∼ U [1 / 3 , 1] 2.1 Problem Compute the Jensen Lower Bound for Q (1 , 3) using the partition Ω = S 1 = { Ω } .
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