Question

Formulate the following problem as least squares problems. For each problem, give a matrixA and a vector b such that the problem can be expressed asargmin)‖Ax − b‖..(you are not asked to solve the problems. Just state define matrix A and vector b)a. minimize x/. + 2x.. + 3x3. + (x/ − x. + x3 − 1). + (−x/ − 4x. + 2).;b. minimize x8x + ||Bx − d||., where the p × n matrix B and the p-vector d are givenc. minimize ||Bx − d||.+2||Fx − g||.. The p × n matrix B, the ???? × n matrix F, the p-vector d and the ????-vector g are given.

Answer 1

a)
`A=\left[\begin{array}{ccc}1&2&3\\1&-1&1\end{array}\right]`

`b=\left[\begin{array}{ccc}0\\1\end{array}\right]`

b)
`||Ax-b||^(2) =(-bx_(2)+4)^(2) (-4x_(1) +3x_(2) -1)^(2) +(x_(1) +8x_(2) -3)^(2)`

c)
`A=\left[\begin{array}{ccc}0&6√(2) &0\\√(3) &3√(3) &0\\2&-16&0\end{array}\right]`

`x=\left[\begin{array}{ccc}x_(1) \\x_(2) \\x_(3) \end{array}\right]`

`b=\left[\begin{array}{ccc}-√(2) \\√(3) \\6\end{array}\right]`

Explanation:

a) considering the equation:

Minimize
`x_(1)^(2) +2x_(2)x^(2) +3x_(3)^(2)+(x_(1) -x_(2) +x_(3) -1)^(2) +(-x_(1) -4x_(2) +2)^(2)`

`A=\left[\begin{array}{ccc}1&2&3\\1&-1&1\end{array}\right]` (matrix A)

vector b

`b=\left[\begin{array}{ccc}0\\1\end{array}\right]`

b) If Pxn is matrix B and p-vector d, we have:

minimize
`(-6x_(2)+4)^(2) +(-4x_(1) +3x_(2) -1)+(x_(1) +8x_(2) -3)^(2)`

`Ax=\left[\begin{array}{ccc}0&-6&0\\-4&3&0\\1&8&0\end{array}\right]`

`\left[\begin{array}{ccc}x_(1) \\x_(2) \\x_(3) \end{array}\right]`

`b=\left[\begin{array}{ccc}-4\\1\\3\end{array}\right]`

`Ax-b=\left[\begin{array}{ccc}-bx_(2)+4 \\-4x_(1)+3x_(2)-1 \\x_(1)+8x_(2)-3 \end{array}\right] =1`

`||Ax-b||^(2) =(-bx_(2)+4)^(2) (-4x_(1) +3x_(2) -1)^(2) +(x_(1) +8x_(2) -3)^(2)`

c) minimize
`2(-bx_(2)+4)^(2) +3(-4x_(1) +3x_(2) -1)^(2) +4(x_(1) -x_(2) -3)^(2) -(6√(2)x_(2) +4√(2) )^(2) +(-4√(3) x_(1) +3√(3)x_(2) -√(3))^(2) +(2x_(1) -16x_(2) -6)^(2)`

in matrix:

`A=\left[\begin{array}{ccc}0&6√(2) &0\\√(3) &3√(3) &0\\2&-16&0\end{array}\right]`

`x=\left[\begin{array}{ccc}x_(1) \\x_(2) \\x_(3) \end{array}\right]`

`b=\left[\begin{array}{ccc}-√(2) \\√(3) \\6\end{array}\right]`