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 de±ned 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 ±nd all the corner points of F . Solution: Like the previous homework, we draw all of the lines and ±nd which side of the lines the feasible region is on, see Figure 1. Then by ±nding 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 ±ve 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 ±nd the constraints whose residuals would decrease after a positive step α along the direction p = (1 , - 2) emanating from the point ¯ x .
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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_hw2_sol - Math 171A Homework 2 Solutions...

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