co227_a5_soln

# co227_a5_soln - Assignment 5 Due Wednesday November 30 at...

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Unformatted text preview: Assignment 5 Due: Wednesday November 30 at the BEGINNING of class 1. Given the following linear programs, write down any 3 Chvatal-Gomory cuts. Explain how you derive each one. (a) maximize 2 x 1 + x 2- x 3 + 3 x 4 + x 5 subject to 3 2 x 1 + 4 3 x 2 + x 4 + 7 3 x 5 ≤ 10 3- x 1 + 12 5 x 2 + 3 5 x 3 + 2 x 5 ≤ 8 5 4 x 2 + 2 3 x 3 + 13 3 x 4 + 7 3 x 5 ≤ 19 6 x 1 ,x 2 ,x 3 ,x 4 ,x 5 ≥ Solution: We can take a CG-cut of each constraint: 3 2 x 1 + 4 3 x 2 + x 4 + 7 3 x 5 ≤ 10 3 ⇒ x 1 + x 2 + x 4 + 2 x 5 ≤ 3- x 1 + 12 5 x 2 + 3 5 x 3 + 2 x 5 ≤ 8 5 ⇒- x 1 + 2 x 2 + 0 x 3 + 2 x 5 ≤ 1 4 x 2 + 2 3 x 3 + 13 3 x 4 + 7 3 x 5 ≤ 19 6 ⇒ 4 x 2 + 0 x 3 + 4 x 4 + 2 x 5 ≤ 3 (b) minimize 19 4 x 1- 5 7 x 2 + 8 3 x 3 + x 4- 1 2 x 5 subject to 2 x 1- 7 3 x 3 + 4 3 x 4- 5 3 x 5 ≤ - 8 3- 3 2 x 1 + 11 2 x 2- 3 2 x 4 + 2 x 5 ≤ - 5 2 1 5 x 1 + x 2- 2 5 x 3 + 17 5 x 4- 11 5 x 5 ≤ - 21 5 x 1 ,x 2 ,x 3 ,x 4 ,x 5 ≥ Solution: We can take a CG-cut of each constraint: 2 x 1 +- 7 3 x 3 + 4 3 x 4 +- 5 3 x 5 ≤ - 8 3 ⇒ 2 x 1- 3 x 3 + x 4- 2 x 5 ≤ - 3- 3 2 x 1 + 11 2 x 2 +- 3 2 x 4 +2 x 5 ≤ - 5 2 ⇒- 2 x 1 +5 x 2- 2 x 4 +2 x 5 ≤ - 3 1 5 x 1 + x 2 + 2 5 x 3 + 17 5 x 4 +- 11 5 x 5 ≤ - 21 5 ⇒ x 1 + x 2- x 3 +3 x 4- 3 x 5 ≤ - 5 (c) maximize 37 x 1 + 23 x 2- 14 x 3 + 7 3 x 4 + 61 13 x 5 subject to 2 x 1- 3 x 2 + x 3 + x 5 ≤ 3- x 1- 7 2 x 2 + 3 2 x 3 + 5 2 x 4 =- 1 2 x 1 + 1 3 x 2 + 4 3 x 3 + 5 3 x 4 + 7 3 x 5 ≥ 10 3 x 1 ,x 2 ,x 3 ,x 4 ,x 5 ≥ Solution: We can take a CG-cut of the second constraint:- x 1 +- 7 2 x 2 + 3 2 x 3 + 5 2 x 4 ≤ - 1 2 ⇒- x 1- 4 x 2 + x 3 +2 x 4 ≤ - 1 We can negate the 3rd constraint (to switch the direction of the inequality) then take a CG-cut:- x 1 +- 1 3 x 2 +- 4 3 x 3 +- 5 3 x 4 +- 7 3 x 5 ≤ - 10 3 ⇒ x 1- x 2- 2 x 3- 2 x 4- 3 x 5 ≤ - 4 We can also take some multiple of the 1st constraint. For taking 1/2 of the 1st constraint then taking a CG-cut: x 1 +- 3 2 x 2 + 1 2 x 3 + 1 2 x 5 ≤ 3 2 ⇒ x 1- 2 x 2 + 0 x 3 + 0 x 5 ≤ 1 2. Consider the following linear programs in canonical form. Note that they each corre- spond to an optimal solution. Find 2 valid cutting planes for each. Explain how youspond to an optimal solution....
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co227_a5_soln - Assignment 5 Due Wednesday November 30 at...

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