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

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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 0 . 5 x 1 , x 2 0 . 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 0 . 5 x 1 , x 2 0 . (b) Consider the problem of minimizing a cost function of the form c 0 x + f ( d 0 · 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 , 0 , 2 x - 4), a piecewise convex linear function. The objective function is c 0 x + f ( d 0 x ) = c 0 x + max( - ( d 0 x ) + 1 , 0 , 2( d 0 x ) - 4). The required formulation is: min c 0 x + z s.t. z ≥ - ( d 0 x ) + 1 z 0 z 2( d 0 x ) - 4 Ax b - 1 2 @ @ @ @ @ @ @ @ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ -1 2 x 1
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Q2. Consider a LP problem in compact form min c 0 x s.t. Ax b . Suppose the LP problem has k optimal solutions x (1) , x (2) , · · · , x ( k ) . Show that k X i =1 λ i x ( i ) is an optimal solution if k X i =1
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