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

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Unformatted text preview: 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 12 will be Problem Set #5, Problems 34 will be Problem Set #6, and Problems 56 will be Problem Set #7. You are allowed to examine outside sources, but you must cite any references that you use. Please dont 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 + 5- 1 0 2 4 x y . Here is a random variable with support = [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 x = (1 , 1 , 1) T . Find the inequality explicitly for this first stage solution x . 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 1 =1 Z 1 2 =1 / 3 y 1 ( 1 , 2 ) + y 2 ( 1 , 2 ) d 1 d 2 subject to 1 x 1 + x 2 + y 1 ( 1 , 2 ) 7 1 , 2 2 x 1 + x 2 + y 2 ( 1 , 2 ) 4 1 , 2 x 1 x 2 y 1 ( 1 , 2 ) 1 , 2 y 2 ( 1 , 2 ) 1 , 2 = { 1 2 } 1 U [1 , 4] 2 U [1 / 3 , 1] 2.1 Problem Compute the Jensen Lower Bound for Q (1 , 3) using the partition = S 1 = { } ....
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hw3 - IE 495 Stochastic Programming Problem Sets #5#7 Due...

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