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math171a_hw5_sol

# math171a_hw5_sol - Math 171A Homework 5 Solutions...

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Math 171A Homework 5 Solutions Instructor: Jiawang Nie February 21, 2012 1. (5 points) Consider the linear program Minimize 3 x 1 - x 2 +2 x 3 subject to - 2 x 1 +4 x 2 +4 x 3 6 x 1 +4 x 2 + x 3 5 - 2 x 1 + x 2 +2 x 3 1 2 x 1 - 2 x 2 0 - 3 x 2 + x 3 ≥ - 2 x 1 0 x 2 0 x 3 0 and the point ¯ x = (1 , 1 , 1) T . Find the active set at ¯ x and determine if the point ¯ x is optimal. If ¯ x is not optimal, ﬁnd a direction p such that c T p < 0 and A a p 0. Solution: Write down the standard form ﬁrst. c = [3 , - 1 , 2] > , A = - 2 4 4 1 4 1 - 2 1 2 2 - 2 0 0 - 3 1 1 0 0 0 1 0 0 0 1 , b = 6 5 1 0 - 2 0 0 0 . Calculate the residual at ¯ x = [1 , 1 , 1] > is r x ) = A ¯ x - b = [0 1 0 0 0 1 1 1] > , so the active set at ¯ x to be { 1 , 3 , 4 , 5 } . To determine if ¯ x is optimal, we must invoke Farka’s Lemma and check the optimality conditions. Farka’s Lemma states that if one has ANY non-negative solution to A T a λ = c , then the point ¯ x is optimal. This amounts to solving A T a λ = c and checking if there are any non-negative solutions. The problem is that A a is not square, and it turns out that A T a λ = c does not have a unique solution. Use the method of calculating the basic solution, we pick the submatrix of A a to get the square submatrix, and solve the problem to get λ , i.e. solve the following problem: A T 134 λ = c, get λ = [0 . 7500 , - 0 . 5000 , 1 . 7500] > A T 135 λ = c, get λ = [1 . 7222 , - 3 . 2222 , 1 . 5556] > 1

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A T 145 λ = c, get λ = [0 . 5714 , 2 . 0714 , - 0 . 2857] > A T 345 λ = c, get λ = [1 . 6000 , 3 . 1000 , - 1 . 2000] > It turns out that we will not be able to ﬁnd a non-negative solution to any of these four problems and thus the point ¯ x is not optimal. To ﬁnd a feasible search direction p , take the solution of the ﬁrst problem. Let s denote the component of λ which is negative, pick the second one, if you then solve A 134 p = e 2 , get p = [ - 0 . 5000 , - 0 . 5000 , 0 . 2500] > . , check this direction p will have the property that c T p = - 0 . 5000 < 0 and we must then check if A a p = [0;1 . 0000;0;1 . 7500] 0, which satisfy the condition.
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math171a_hw5_sol - Math 171A Homework 5 Solutions...

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