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Unformatted text preview: IEOR 160: Midterm Question Solutions Question 1 Solution a) Assume there are x people drive. The nonlinear programming formulation is as follows: min 0 . 005 x 2 20 x + 400 , 000 s.t. x 10 , 000 b) The KKT conditions for the above NLP are: (0 . 01 x 20) +  = 0 (1) (10 , 000 x ) = 0 (2) (0 . 01 x 20 + ) x = 0 (3) , (4) c) Solve the KKT conditions, first consider equation (3), if x = 0, then from (1), 20 0, this is impossible for x 0. Thus, x 6 = 0. Then, = 20 . 01 z . Plug in to (2), get (20 . 01 x )( x 10 , 000) = 0, from which we get x = 10 , 000 or x = 2 , 000. If x = 10 , 000, < 0. Thus, the only solution to the KKT conditions is ( x, ) = (2 , 000 , 0) d) The solution of the KKT conditions from part c) give an optimal solution for the NLP, since the objective function has f 00 ( x ) = 0 . 01 thus is convex and the constraints are linear, thus convex as well, also the constraints qualification always satisfies. Therefore, the KKT solutionconvex as well, also the constraints qualification always satisfies....
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 Spring '07
 HOCHBAUM

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