June 17th, 2017, 04:17 AM  #1 
Newbie Joined: Jun 2017 From: Israel Posts: 1 Thanks: 0  Least Squares
Hi, I got this question but I don't know how to solve it: Consider the following optimization problem: sqrt((x11)^2+(2*x2+3)^2+x1^2+x2^2x1*x2) Write down the above problem as the least squares problem and solve it. Last edited by skipjack; June 17th, 2017 at 06:29 AM. 
June 17th, 2017, 07:49 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,422 Thanks: 1462 
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 $\sqrt{106/39}$. 
June 17th, 2017, 08:00 AM  #3  
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  Quote:
$\displaystyle \sqrt{2x_1^2 + 2x_1 + 5x_2^2 + 12x_2 x_1x_2 + 10}$ 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 08:04 AM.  

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