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# practicefinal - OR3300/5300 Optimization I Final exam...

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OR3300/5300 Optimization I 12/4/2011 Final exam Practice problems on the material since prelim 2. 1. (30 points total) Consider the linear programming problem maximize 5 x 1 + 0 x 2 + 0 x 3 + 10 x 4 s.t. 1 x 1 + 1 x 2 + 1 x 3 + 1 x 4 4 - 1 x 1 + 2 x 2 + 3 x 3 + 3 x 4 1 ( P ) 8 x 1 - 6 x 2 + 7 x 3 + 2 x 4 2 9 x 1 + 10 x 2 - 11 x 3 + 4 x 4 20 x 1 0 , x 2 0 , x 3 0 , x 4 0 . (a) (5 points) Write down the linear programming dual of (P). (Let y i be the dual variable associated with the ith less-than-or-equal-to constraint of (P).) (b) (5 points) Write down the complementary slackness conditions for (P) and its dual.

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(c) (5 points) Explain why the vectors x = (4 , - 1 , 0 , 0) T and y = (0 , 0 , 0 , 1) are not complementary for this pair of problems (P) and (D). (d) (15 points) You need to check whether the feasible solution for (P) given by x 1 = 1 , x 2 = 1 , x 3 = 0 , x 4 = 0 is optimal. Use your answers to (a) and (b) to determine this; do not use the simplex method. State any results that you are using.

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2. (30 points total) We are solving a linear programming problem of the form maximize cx subject to Ax b x
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practicefinal - OR3300/5300 Optimization I Final exam...

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