IEOR 160-practice midterm-1

IEOR 160-practice midterm-1 - i=1 2 3 Write the optimality...

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IEOR 160 - Practice Midterm Fall 2009 Problem 1 A monopolist producing a single product has two customers. If q 1 units are produced for customer 1, then customer 1 is willing to pay a price of 70 – 4q 1 dollars per unit. If q 2 units are produced for customer 2, then customer 2 is willing to pay a price of 150 – 17q 2 dollars per unit. For q > 0, the cost of manufacturing a total of q units is 100 + 14q dollars. Formulate an unconstrained non-linear program to help the monopolist decide how much she should sell to each customer in order to maximize the total profit. Solve the problem. Problem 2 Consider the following problem: min f(x 1 ,x 2 )=100(x 2 -x 1 2 ) 2 +(1-x 1 ) 2 a) Find all local minimum points of f. b) Which of the points found in a) are global minimum points? Justify your answers. Problem 3 Consider the problem: Min x 1 2 + x 2 2 + x 3 2 -x 1 x 2 - 2x 3 subject to x 1 +x 2 +x 3 ≤ 0.5 x i ≥ 0
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Unformatted text preview: i=1, 2, 3 Write the optimality conditions for the problem and use it to obtain the optimal solution. Justify your answer. Problem 4 Read each of the following statements carefully to see whether it is true or false . Justify your answers (no credit for answers without justification!) 1) If x* is a point satisfying the Karush-Kuhn-Tucker conditions for a maximization problem with a concave objective function over some inequality constrains then x* is a global maximum point. 2) If all leading principal minors of the Hessian of a function f : Rn→R are positive for all points in Rn, then f is a convex function. 3) For the following single variable nonlinear programming problem: Max f(x) s.t. g(x)=b, Let L(x,λ)=f(x)+λ(b-g(x)). If (x*,λ*) satisfies: ∂f/∂x=∂f/∂λ=0, then x* is an optimal solution....
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IEOR 160-practice midterm-1 - i=1 2 3 Write the optimality...

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