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Unformatted text preview: IEOR 160 Practice Midterm Question Solutions Question 1 Solution The nonlinear programming formulation is max (70 4 q 1 ) q 1 + (150 17 q 2 ) q 2 (100 + 14 q 1 + 14 q 2 ) s.t. q 1 ≥ q 2 ≥ The Hessian of the objective function H ( x ) = 8 34 ¶ Since the Hessian of the objective function is a negative definite matrix, thus, the objective func tion is concave. Therefore, the KKT conditions for the above NLP are sufficient for optimality. the point such that firstorder derivative is zero is given by 8 q 1 + 56 = 0 34 q 2 + 136 = 0 We get q * 1 = 7 ≥ 0, q * 2 = 4 ≥ 0. Therefore, the monopolist will produce 7 units for the first customer and produce 4 units for the second customer. Question 2 Solution Solve ∇ f ( x ) = 0 for x, we can get 200( x 2 x 2 1 )( 2 x 1 ) + 2(1 x 1 )( 1) = 0 200( x 2 x 2 1 ) = 0 We get x * 1 = 1 , x * 2 = 1. The Hessian of the objective function is H ( x ) = 1200 x 2 1 400 x 2 + 2 400 x 1 400 x 1 200 ¶ The Hessian is not a positive definite matrix for all the...
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This note was uploaded on 02/17/2010 for the course IEOR 160 taught by Professor Hochbaum during the Fall '07 term at Berkeley.
 Fall '07
 HOCHBAUM
 Operations Research

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