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ISYE 3133 Modeling3

# ISYE 3133 Modeling3 - min m,c max i =1 4 | mx i c-y i |(2 1...

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ISYE3133C - Engineering optimization Modeling Handout #3 A quantity y is known to depend upon another quantity x . A set of corresponding values has been collected for x and y and is presented in the following table. i 1 2 3 4 x i 0 . 0 0 . 5 1 . 0 1 . 5 y i 1 . 0 0 . 9 0 . 7 1 . 5 We want to approximate the relation between x and y by fitting the best straight line y = mx + c through the points ( x i , y i ) in the table. To achieve this we need to find parameters m and c . For a particular choice of parameters we have that δ i = | mx i + c - y i | is the error of the approximation for pair ( x i , y i ). For a perfect approximation we would have δ i = 0 for all i . One option is to find a line that minimizes the sum of errors δ i . This is achieved by solving min m,c 4 i =1 | mx i + c - y i | . (1) Another option is to find a line that minimizes the maximum of errors δ i . This is achieved by solving
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Unformatted text preview: min m,c max i =1 ,..., 4 | mx i + c-y i | . (2) 1. Formulate a linear programming problem to ﬁnd m and c which minimize the sum of the errors δ i . 2. Formulate a linear programming problem to ﬁnd m and c which minimize the maximum of the errors δ i . 3. Suppose we now want to ﬁt the best quadratic line y = ax 2 + bx + c . Formulate a linear programming problem to ﬁnd a , b and c which minimize the sum of the errors | ax 2 i + bx i + c-y | . 4. Suppose we now want to ﬁt the best quadratic line y = ax 2 + bx + c . Formulate a linear programming problem to ﬁnd a , b and c which minimize the maximum of the errors | ax 2 i + bx i + c-y | . 1...
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