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math171a_hw2_sol

# math171a_hw2_sol - Math 171A Homework 2 Solutions...

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Math 171A Homework 2 Solutions Instructor: Jiawang Nie February 1, 2012 1. (8 points) Consider the feasible set F defined by the following constraints x 1 + x 2 4 , x 1 + 3 x 2 6 , 6 x 1 - x 2 18 , 3 x 2 6 , x 1 ≥ - 1 . (a) Express F in the standard form Ax b . Write down A and b explicitly. Solution: A = 1 1 1 3 - 6 1 0 1 0 - 1 1 0 b = 4 6 - 18 3 - 6 - 1 (b) Draw the set F graphically in the plane, and find all the corner points of F . Solution: Like the previous homework, we draw all of the lines and find which side of the lines the feasible region is on, see Figure 1. Then by finding which lines intersect we can solve for the corners. The corners that lie in the feasible region should be ( - 1 , 5) , (1 , 3) , (3 . 5 , 3) , (4 , 6) , ( - 1 , 6) (c) Solve the following LP minimize 2 x 1 - 3 x 2 subject to Ax b where A and b are from part (a). Solution: Since we know the corner points of the feasible region from part (b), we need to plug those points into the objective function and compare. At five corner points, the objective function values are -17, -7, -2, -10, -20 respectively. The point which minimizes 2 x 1 - 3 x 2 is the point ( - 1 , 6) with a value of - 20. 1

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-1 0 1 2 3 4 5 6 -25 -20 -15 -10 -5 0 5 10 15 20 C6 Feasible Region C5 C4 C3 C2 C1 Figure 1: Feasible Region of Question 1. (d) Compute the residual vector r ( x ) for all the constraints at the point ¯ x = (2 , 4), and find the constraints whose residuals would decrease after a positive step α
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