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# sol1 - 2-1 6 1 3-1 2 ∼ 2-1 6 2 9-1 2 ∼ 2-1 6 13-1 2(2...

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CO350 L INEAR O PTIMIZATION - S OLUTION HW1 Exercises 1.6.1 - x 1 6 x 3 0 ¡ ¡ ¡ ¡ ¡ ¡ ¡“ x 2 s 24 s 12 s 6 » » » » » » » » » » » » » » X X X X X X X X X X ←- feasible region The feasible region is outlined in the above picture. The geometric method for solving linear programs tells us that we can find an optimal solution among the extreme points (the ”corner points”). We try (24 , 0 , 0) , (0 , 12 , 0) , and (0 , 0 , 6) and find out that (24 , 0 , 0) gives the maximal value of 72 . Hence, (24 , 0 , 0) is the optimal solution. Ex 1. The LP has a single variable x , min y s.t. y a i ( i = 1 , . . . , k ) Ex 2. (1) The last 3 columns of M are linearly independent. To prove this, transform the matrix by doing elementary row operations (this does not change linear dependence of the columns) and note that the final matrix is (after permutation of the rows and columns) a diagonal matrix with all diagonal elements distinct from
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Unformatted text preview: 2-1 6 1 3-1 2 ∼ 2-1 6 2 9-1 2 ∼ 2-1 6 13-1 2 (2) For any matrix M , the rank of the row vectors is equal of the rank of the column vectors. Hence, if M has only 3 rows there are at most 3 linearly independent columns. Ex 3. (1) For any feasible solution [ x 1 x 2 x 3 ] T ,-x 1 + x 2 ≤ -x 1 + 2 x 2 = ( x 1 + x 2 + x 3 )-(2 x 1-x 2 + x 3 ) ≤ where the ﬁrst inequality follows from the fact that x 2 ≥ and the second one from the fact that x 1 + x 2 + x 3 ≤ 1 and that 2 x 1-x 2 + x 3 = 1 . (2) The vector [0 0 1] T is a feasible solution with objective value . It follows from part (1) that it is an optimal solution. 1...
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