HW1 solution - Solution of Homework 1 Problem(1 Solution We...

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Solution of Homework 1 Problem (1): Solution: We want to find the coefficients x 1 , x 2 , and x 3 such that b (0) = 1 , b (2)= 7 , and b (5)=46 . These three equations can be written in a matrix form: 1 0 0 1 2 4 1 5 25 x 1 x 2 x 3 = 1 7 46 The solution is 1 1 2 squaresolid Problem (2): Solution: Let x 1 be the number of desks, x 2 the number of bookshelves, x 3 the number of cabinets with door, and x 4 the number of cabinets with drawers. Then the optimization can be formulated as Maximize 200 x 1 +100 x 2 +150 x 3 +200 x 4 subject to 8 x 1 +6 x 2 +2 x 3 +4 x 4 2000 12 x 1 +10 x 2 +25 x 3 +20 x 4 6000 x 1 ,x 2 ,x 3 ,x 4 0 squaresolid 1
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Problem (3) Solution: This is similar to problem (1). We want to find the coefficients x 1 and x 2 such that b (0) = 3 , b (1) = 3 , and b (1) = 6 . These can be written in a matrix form: Ax = b with A = 1 0 1 1 1 1 and b = 3 3 6 But there is no solution for this system of equations. In this case, we formulate it as a optimization problem by minimizing the error r = b Ax in the sense of L 2 norm (Euclidean distance). Minimize F ( x )= 1 2 || b Ax || 2 This is a quadratic function and the minimizer can be found by solving
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