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Tutorial7sol - Tutorial 7 Outline of Solutions Q1 Consider...

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Tutorial 7: Outline of Solutions Q1. Consider the following linear programming problem: ( P ) min 3 x 1 + 4 x 3 s.t. 2 x 1 + x 2 - x 3 - 2 x 1 + 3 x 2 - 5 x 3 7 x 1 0 , x 2 0 (a) Write the standard form of the above LP problem. (b) Verify that the dual of the above LP problem obtained directly from the table of primal- dual relation given in the lecture and the dual of the standard form LP problem in part (a) are equivalent. Solution : (a) The standard form LP: ( P 1) min - x 1 + 4 x + 3 - 4 x - 3 s.t. - x 1 + x 2 - x + 3 + x - 3 + s 1 = - 2 - ¯ x 1 + 3 x 2 - 5 x + 3 + 5 x - 3 - s 2 = 7 ¯ x 1 , x 2 , x + 3 , x - 3 , s 1 , s 2 0 (b) Dual (D) of given LP: ( D ) max - 2 p 1 + 7 p 2 s.t. 2 p 1 + p 2 3 p 1 + 3 p 2 0 - p 1 - 5 p 2 = 4 p 1 0 , p 2 0 Dual of standard form LP: ( D 1) max - 2 q 1 + 7 q 2 s.t. - 2 q 1 - q 2 - 3 q 1 + 3 q 2 0 - q 1 - 5 q 2 4 q 1 + 5 q 2 - 4 q 1 0 - q 2 0 q 1 free , q 2 free This is equivalent to ( D 1) max - 2 q 1 + 7 q 2 s.t. - 2 q 1 - q 2 - 3 q 1 + 3 q 2 0 - q 1 - 5 q 2 = 4 q 1 0 q 2 0 1
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which is the same as (D) with p 1 = q 1 , and p 2 = q 2 . Q2. (Consider the following linear programming problem (P) ( P ) min c 0 x s.t. Ax b x 0 (a) Write down the dual problem ( D ).
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