Tut2sol - Tutorial 2 Outline of Solutions Q1(a Reformulate...

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Unformatted text preview: Tutorial 2: Outline of Solutions Q1. (a) Reformulate the following problem as a linear programming problem: max min( x 1 ,x 2 ) s.t. | 2 x 1 + x 2 | ≤ 7 3 x 1- x 2 1 + x 1 + x 2 ≤ . 5 x 1 ,x 2 ≥ . Solution : max z s.t. z ≤ x 1 z ≤ x 2 2 x 1 + x 2 ≤ 7- 2 x 1- x 2 ≤ 7 2 . 5 x 1- 1 . 5 x 2 ≤ . 5 x 1 ,x 2 ≥ . (b) Consider the problem of minimizing a cost function of the form c x + f ( d · x ), subject to the linear constraints Ax ≥ b . Here d is a given vector and the function f : R → R is as specified in the following figure. Provide a linear programming formulation of this problem. Solution : Note f ( x ) = max(- x +1 , , 2 x- 4), a piecewise convex linear function. The objective function is c x + f ( d x ) = c x + max(- ( d x ) + 1 , , 2( d x )- 4). The required formulation is: min c x + z s.t. z ≥ - ( d x ) + 1 z ≥ z ≥ 2( d x )- 4 Ax ≥ b- 1 2 @ @ @ @ @ @ @ @ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢-1 2 x 1 Q2. Consider a LP problem in compact form min c x s.t. Ax ≥ b ....
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This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.

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Tut2sol - Tutorial 2 Outline of Solutions Q1(a Reformulate...

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