Practice Midterm Solution revised

Practice Midterm Solution revised - IEOR 160 Practice...

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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 first-order 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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Practice Midterm Solution revised - IEOR 160 Practice...

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