HW5_Sol - E160 Operations Research I Spring 2009 Homework...

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E160 Operations Research I Spring 2009 Homework #5 Solution Chapter 12 Section 12.6 1. We wish to minimize f(x, y) = Σ {(x i - x) 2 + (y i - y) 2 } i = 1 i = n f/ x = -2 Σ (x i - x) = 0 for x* = (x 1 + x 2 +... +x n )/n i = 1 i = n f/ y= -2 Σ (y i - x) = 0 for y* = (y 1 + y 2 +... +y n )/n i = 1 f(x,y) is the sum of convex functions and is therefore convex. To show that (x*, y*) is a local minimum we compute the Hessian. = n n H 2 0 0 2 Thus H 1 = 2n >0 and H 2 = 4n 2 >0, and (x*, y*) is a local minimum. The fact that f(x,y) is convex now shows that (x*, y*) minimizes f(x,y) over all possible choices of (x, y). 2. We wish to maximize f(q 1 , q 2 ) = 2(q 1 1/3 + q 2 2/3 ) - q 1 - 1.5q 2 f/ q 1 = 2q 1 -2/3 /3 - 1 = 0 for q 1 = (2/3) 3/2 f/ q 2 = 4q 2 -1/3 /3 - 1.5 = 0 for q 2 = (4/4.5) 3 The objective function is the sum of concave functions, so we know that the above values for q 1 and q 2 will maximize profit. 3. Let f(q
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HW5_Sol - E160 Operations Research I Spring 2009 Homework...

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