hw2 - IE426 Problem Set#2 Prof Jeff Linderoth IE 426...

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IE426 Problem Set #2 Prof Jeff Linderoth IE 426 – Problem Set #2 Due Date: September 26, 2006. 4:30PM. Note: No late homework will be accepted, as I will be discussing solutions on 9/26 1 LP Knowledge For each of the problems in this section, you should determine which of the characterizations below best describes the linear program. The linear program may fall into more than one category, in which case write all characterizations that apply. A Unique Optimal B Multiple Optima C Infeasible D Unbounded E Degenerate 1.1 Problem min - x 1 - x 2 s.t. 6 x 1 + 2 x 2 8 - x 1 + x 2 0 x 1 0 x 2 0 1.2 Problem max 3 x 1 + x 2 s.t. 6 x 1 + 2 x 2 8 - x 1 + x 2 0 x 1 0 x 2 0 1.3 Problem min x 2 Problem 1 Page 1
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IE426 Problem Set #2 Prof Jeff Linderoth s.t. x 1 1 6 x 1 + 2 x 2 8 - x 1 + x 2 0 x 1 0 x 2 0 2 Least “Squares.” Warning. It is likely that you will need the XPRESS keyword is free . The set of six equations in four variables (1)—(6) does not have a unique solution. 1 8 x 1 - 2 x 2 + 4 x 3 - 9 x 4 = 17 (1) x 1 + 6 x 2 - x 3 - 5 x 4 = 16 (2) x 1 - x 2 + x 3 = 7 (3) x 1 + 2 x 2 - 7 x 3 + 4 x 4 = 15 (4) x 3 - x 4 = 6 (5) x 1 + x 3 - x 4 = 0 (6) For each equation i , and values of variables x = ( x 1 , x 2 , x 3 , x 4 ) , let e i be the absolute difference (error) between the left hand side and the right hand side. For example, for i = 2 and x = ( - 5 , 3 , 1 , 4) , the error is e 2 = | (1)( - 5) + 6(3) - (1)(1) - 5(4) - 16 | = | - 24 | = 24 . 2.1 Problem Write a linear programming instance that will minimize the total absolute error: e 1 + e 2 + e 3 + e 4 + e 5 + e 6 2.2 Problem Create the instance you wrote down for Problem 2.1 in the Mosel modeling language, and solve it. What is the minimum total error that can be achieved? What are the values for x ? 2.3 Problem For the same instance, write a linear programming instance that will minimize the maxi- mum error in any one equation. Namely find values of x that will min max { e 1 , e 2 , e 3 , e 4 , e 5 , e 6 } for the definition of error e i I gave above.
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