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June 17th, 2017, 03:17 AM   #1
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Least Squares

Hi,

I got this question but I don't know how to solve it:

Consider the following optimization problem:

sqrt((x1-1)^2+(2*x2+3)^2+x1^2+x2^2-x1*x2)

Write down the above problem as the least squares problem and solve it.

Last edited by skipjack; June 17th, 2017 at 05:29 AM.
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June 17th, 2017, 06:49 AM   #2
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The image contains $\displaystyle \min_{(x_1,x_2)\in\mathbb{R}^2}\!\!\sqrt{(x_1 - 1)^2 + (2x_2 + 3)^2 + x_1^2 + x_2^2 - x_1x_2}$.

$(x_1 - 1)^2 + (2x_2 + 3)^2 + x_1^2 + x_2^2 - x_1x_2 = \frac18((4x_1 - x_2 - 2)^2 + 39(x_2 + \frac{46}{39})^2) + \frac{106}{39}$,
which has a minimum value of 106/39 when $x_1 = 8/39$ and $x_2 = -46/39$.

Hence the desired minimum value is $\sqrt{106/39}$.
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June 17th, 2017, 07:00 AM   #3
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Quote:
Originally Posted by Kereny View Post
Hi,

I got this question but I don't know how to solve it:

Consider the following optimization problem:

sqrt((x1-1)^2+(2*x2+3)^2+x1^2+x2^2-x1*x2)

Write down the above problem as the least squares problem and solve it.
So, the first thing I did was to simply expand everything and get rid of those pesky brackets:

$\displaystyle \sqrt{2x_1^2 + 2x_1 + 5x_2^2 + 12x_2 -x_1x_2 + 10}$

The most annoying thing here, I think, is the 'interaction term' $\displaystyle x_1x_2$. Let's stuff it into a squared expression:

$\displaystyle \sqrt{2x_1^2 + 2x_1 + 5x_2^2 + 12x_2 +(\frac{1}{4} x_1^2 - x_1x_2 + x_2^2) - \frac{1}{4}x_1^2 - x_2^2 + 10}$


now we can simplify:

$\displaystyle \sqrt{\frac{7}{4}x_1^2 + 2x_1 + 4x_2^2 + 12x_2 + (\frac{1}{2} x_1 - x_2)^2 + 10}$

and now, by completing the squares, we have:

$\displaystyle \sqrt{\frac{7}{4}(x_1+\frac{4}{7})^2 - \frac{4}{7}+4(x_2+\frac{3}{2})^2 - 9 + (\frac{1}{2} x_1 - x_2)^2 + 10}$

Then, of course, we ignore the little constant and the square root because we're minimising and YOLO:

$\displaystyle {(\frac{\sqrt 7}{2}x_1+\frac{2}{\sqrt 7})^2 +(2x_2+3)^2 + (\frac{1}{2} x_1 - x_2)^2 }$

And of course this is equivalent to

$\displaystyle \left \|\begin{bmatrix} \frac{\sqrt 7}{2}x_1+\frac{2}{\sqrt 7}\\ 2x_2+3\\\frac{1}{2} x_1 - x_2 \end{bmatrix} \right \|^2$

i.e.

$\displaystyle \left \|\begin{bmatrix} \frac{\sqrt 7}{2} & 0\\0 & 2\\ \frac{1}{2} & -1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} - \begin{bmatrix} -\frac{2}{\sqrt 7}\\-3 \\ 0\end{bmatrix} \right \|^2$

This is tedious though haha. Probably not the best way to do it - if the question didn't specify a method, I'd just differentiate. (And as usual, I apologise in advance for any mistakes here. I'd actually be surprised if I didn't make any...)

Last edited by 123qwerty; June 17th, 2017 at 07:04 AM.
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