midterm

# midterm - NAME PAGE 2 OF 8 Question 1[5 marks Convert the...

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Unformatted text preview: NAME: PAGE 2 OF 8 Question 1. [5 marks] Convert the following linear program into standard equality form. min 3 x 1- 2 x 2 + x 3 subject to 2 x 1- x 2 ≥ 2 3 x 2- 5 x 3 = 6- x 2 +3 x 3 ≤ - 3 x 1 ≤ ,x 3 ≥ Solution: max 3 x 1 + 2 x + 2- 2 x- 2- x 3 subject to- 2 x 1- x + 2 + x- 2- s 1 = 2 3 x + 2- 3 x- 2- 5 x 3 = 6- x + 2 + x- 2 + 3 x 3 + s 2 =- 3 x 1 , x + 2 , x- 2 , x 3 , s 1 , s 2 ≥ . NAME: PAGE 3 OF 8 Question 2. [10 marks] Consider the linear program (P) max x 1 + x 2 subject to x 1 + 3 x 2- 6 x 3 + x 4 = 5 + x 2- 3 x 3- x 4 = 0 x 1- x 2 + 6 x 3 + 5 x 4 = 5 x 1 , x 2 , x 3 , x 4 , ≥ . (a) Write the dual (D) of (P). Solution: min 5 y 1 + 5 y 3 subject to y 1 + y 3 ≥ 1 3 y 1 + y 2- y 3 ≥ 1- 6 y 1- 3 y 2 + 6 y 3 ≥ y 1- y 2 + 5 y 3 ≥ (b) State the complementary slackness conditions for (P) and (D). Solution: For feasible solutions x and y of (P) and (D), respectively, x and y are optimal if and only if each of the following conditions hold: x 1 = 0 or y 1 + y 3 = 1 x 2 = 0 or 3 y 1 + y 2- y 3 = 1 x 3 = 0 or- 6 y 1- 3 y 2 + 6 y 3 = x 4 = 0 or y 1- y 2 + 5 y 3 = 0 . (c) Is the feasible solution x = (2 , 3 , 1 , 0) T of (P) optimal? Solution: x is optimal if and only if there exists a feasible solution y for (D) satisfying the com- plementary slackness conditions with x . So x is optimal if and only if the following linear system...
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midterm - NAME PAGE 2 OF 8 Question 1[5 marks Convert the...

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