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Answers to HW due on Oct 14

# Answers to HW due on Oct 14 - Answers to assignment due on...

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Answers to assignment due on October 14 th . 2. Program 1D: * z = min y b, subject to y A c . To place this thing in the format of Program 12.1, we need to express it as a maximization problem with nonnegative variables and equality constraints. Let’s replace y by the difference yp - yn of two vectors of nonnegative variables, let’s insert a vector of surplus variables to convert the inequalities into equations, and let’s multiply the objective by -1. This gives - * z = max (yn – yp) b , subject to v (yp – yn) A – w = c , yp 0 , yn 0 , w 0 . - Taking the dual of the latter gives - * z = min (c v) , subject to yn: - A v b , yp: A v ≥ – b , w: - v 0 . Substituting x = - v and switching the sense of the objective gives * z = max c x , subject to A x = b , x 0 , which is Program 12.1 3. Denote as j x the number of units of food j that the consumer consumes. (a) The minimum-cost diet that satisfies his nutritional needs is found by solving the linear program: Minimize = n 1 j j j x c , subject to the constraints i y = n 1 j i j ij b

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