practice_midterm_2

# practice_midterm_2 - Math 171A Practice Midterm II...

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Math 171A Practice Midterm II Instructor: Jiawang Nie March 5, 2012 1. Consider the LP of minimizing c > x subject to Ax b where A = 2 3 5 3 4 3 4 5 1 2 4 4 3 5 2 , b = - 15 - 13 - 20 - 12 - 13 ,c = A > 0 1 0 1 1 . Find the minimizer of this LP by using optimality conditions (do not use simplex method). Also explain why your solution is a minimizer. Solution: Pick A a = A { 2 , 4 , 5 } , and let λ = [1 , 1 , 1] > 0, c = A > a λ . Solve A a x = b { 2 , 4 , 5 } , x * = [ - 2 , - 1 , - 1] > . By Farkas Lemma, this x * is a minimizer. 2. Let a = ( a 1 ,...,a n ) R n + be a positive vector. Consider the LP: Maximize x 1 + x 2 + ··· + x n subject to a 1 x 1 + a 2 x 2 + ··· + a n x n 1 x 1 0 ,...,x n 0 . Find the solution x * of the above LP by using optimality conditions (i.e., do not use the simplex method), and express it in terms of a i . Explain why your solution is optimal.

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practice_midterm_2 - Math 171A Practice Midterm II...

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