math171a_hw4_sol

math171a_hw4_sol - Math 171A Homework 4 Solutions...

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Unformatted text preview: Math 171A Homework 4 Solutions Instructor: Jiawang Nie February 14, 2012 1. (4 points) Consider the matrix of active constraints of an LP at a certain point ¯ x A a (¯ x ) = - 1- 8 4- 7- 3- 4 2 4- 2- 6- 3 3 5 5 2 7 . Use Matlab to find a direction p along which the residual of the third constraint increases, but all the other residuals remain the same. If the rows of A a (¯ x ) are labeled 1 through 4, what are the indices of the active set after a positive step along this direction? Solution: By definition of residual, r ( x ) = A ( x )- b , start from ¯ x , which has r a (¯ x ) = A a (¯ x )- b a = 0. Go along p with positive step, new residual r (¯ x + αp ) = r (¯ x ) + αA a p . To make the third one increase and the rest keep the same, we just need to solve A a p = e 3 . There are many solutions, choose column { 1 , 2 , 3 , 4 } , get one direction p = [0 . 2741;- . 7111;- . 2519; 0 . 6296] > . After a positive step along this direction, the active set is { 1 , 2 , 4 } . 2. (5 points) Let F be the feasible set defined as F = { x ∈ R 2 : | x 1 | + 2 | x 2 | ≤ 2 , | x 1 | ≤ 1 , | x 2 | ≤...
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This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.

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math171a_hw4_sol - Math 171A Homework 4 Solutions...

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