Finding $(x,y)$ closest to $N$ points
$begingroup$
Given $N$ points where the coordinates of the $i$th point are $(x_i,y_i)$ find $(x,y)$ that minimizes the following sum.
$$sum_{i=1}^ Nmax(|x-x_i|,|y-y_i|)$$
I've done a similar problem minimize
$$sum_{i=1}^ N(|x-x_i|+|y-y_i|)$$
Which I found that the minimum is achieved when we take $x$ as the median of $x_1,x_2,ldots,x_N$ and $y$ as median of $y_1,y_2,ldots,y_N$.
I'm stuck on the first one because I don't know how to manipulate the $max$.
Edit 1:
After tinkering with the suggestion below I am still struggling to make progress here: The $frac{|x-x_i|+|y-y_i|}{2}$ can be minimized by taking the median but I am not sure how to minimize the other part: $frac{||x|-|x_i||-||y|-|y_i||}{2}$. I also don't know what to do if I could minimize the other part. How can I minimize the sum of these two?
optimization algorithms
edited Dec 23 '18 at 16:26
Mason
1,9751530
asked Nov 30 '18 at 17:36
kingW3kingW3
11k72555
$endgroup$
add a comment |
$begingroup$
Given $N$ points where the coordinates of the $i$th point are $(x_i,y_i)$ find $(x,y)$ that minimizes the following sum.
$$sum_{i=1}^ Nmax(|x-x_i|,|y-y_i|)$$
I've done a similar problem minimize
$$sum_{i=1}^ N(|x-x_i|+|y-y_i|)$$
Which I found that the minimum is achieved when we take $x$ as the median of $x_1,x_2,ldots,x_N$ and $y$ as median of $y_1,y_2,ldots,y_N$.
I'm stuck on the first one because I don't know how to manipulate the $max$.
Edit 1:
After tinkering with the suggestion below I am still struggling to make progress here: The $frac{|x-x_i|+|y-y_i|}{2}$ can be minimized by taking the median but I am not sure how to minimize the other part: $frac{||x|-|x_i||-||y|-|y_i||}{2}$. I also don't know what to do if I could minimize the other part. How can I minimize the sum of these two?
optimization algorithms
edited Dec 23 '18 at 16:26
Mason
1,9751530
asked Nov 30 '18 at 17:36
kingW3kingW3
11k72555
$endgroup$
1
$begingroup$
Not a complete answer which finds an expression for the $(x,y)$. However, here is an identity that I think will probably get you where you want to go. $max{{a,b}} = frac{a+b}{2} + frac{|a-b|}{2}$
$endgroup$
– Mason
Dec 23 '18 at 15:55
$begingroup$
I hope I didn't rob you of more complete answers by putting something more worthy as a comment as an answer. Anyway: I put a suggested edit and deleted my answer to get this back on the list of unanswered questions+ an edit will bump this back on to active questions.
$endgroup$
– Mason
Dec 23 '18 at 16:06
$begingroup$
@Mason Nice of you to do that, I don't think you robbed me of an answer. I feel that the question isn't interesting to people and your observation is helpful.
$endgroup$
– kingW3
Dec 23 '18 at 16:36
add a comment |
$begingroup$
Given $N$ points where the coordinates of the $i$th point are $(x_i,y_i)$ find $(x,y)$ that minimizes the following sum.
$$sum_{i=1}^ Nmax(|x-x_i|,|y-y_i|)$$
I've done a similar problem minimize
$$sum_{i=1}^ N(|x-x_i|+|y-y_i|)$$
Which I found that the minimum is achieved when we take $x$ as the median of $x_1,x_2,ldots,x_N$ and $y$ as median of $y_1,y_2,ldots,y_N$.
I'm stuck on the first one because I don't know how to manipulate the $max$.
Edit 1:
After tinkering with the suggestion below I am still struggling to make progress here: The $frac{|x-x_i|+|y-y_i|}{2}$ can be minimized by taking the median but I am not sure how to minimize the other part: $frac{||x|-|x_i||-||y|-|y_i||}{2}$. I also don't know what to do if I could minimize the other part. How can I minimize the sum of these two?
optimization algorithms
edited Dec 23 '18 at 16:26
Mason
1,9751530
asked Nov 30 '18 at 17:36
kingW3kingW3
11k72555
$endgroup$
Given $N$ points where the coordinates of the $i$th point are $(x_i,y_i)$ find $(x,y)$ that minimizes the following sum.
$$sum_{i=1}^ Nmax(|x-x_i|,|y-y_i|)$$
I've done a similar problem minimize
$$sum_{i=1}^ N(|x-x_i|+|y-y_i|)$$
Which I found that the minimum is achieved when we take $x$ as the median of $x_1,x_2,ldots,x_N$ and $y$ as median of $y_1,y_2,ldots,y_N$.
I'm stuck on the first one because I don't know how to manipulate the $max$.
Edit 1:
After tinkering with the suggestion below I am still struggling to make progress here: The $frac{|x-x_i|+|y-y_i|}{2}$ can be minimized by taking the median but I am not sure how to minimize the other part: $frac{||x|-|x_i||-||y|-|y_i||}{2}$. I also don't know what to do if I could minimize the other part. How can I minimize the sum of these two?
optimization algorithms
optimization algorithms
edited Dec 23 '18 at 16:26
Mason
1,9751530
asked Nov 30 '18 at 17:36
kingW3kingW3
11k72555
edited Dec 23 '18 at 16:26
Mason
1,9751530
asked Nov 30 '18 at 17:36
kingW3kingW3
11k72555
edited Dec 23 '18 at 16:26
Mason
1,9751530
edited Dec 23 '18 at 16:26
Mason
1,9751530
edited Dec 23 '18 at 16:26
Mason
1,9751530
1,9751530
asked Nov 30 '18 at 17:36
kingW3kingW3
11k72555
asked Nov 30 '18 at 17:36
kingW3kingW3
11k72555
asked Nov 30 '18 at 17:36
kingW3kingW3
11k72555
11k72555
1
$begingroup$
Not a complete answer which finds an expression for the $(x,y)$. However, here is an identity that I think will probably get you where you want to go. $max{{a,b}} = frac{a+b}{2} + frac{|a-b|}{2}$
$endgroup$
– Mason
Dec 23 '18 at 15:55
$begingroup$
I hope I didn't rob you of more complete answers by putting something more worthy as a comment as an answer. Anyway: I put a suggested edit and deleted my answer to get this back on the list of unanswered questions+ an edit will bump this back on to active questions.
$endgroup$
– Mason
Dec 23 '18 at 16:06
$begingroup$
@Mason Nice of you to do that, I don't think you robbed me of an answer. I feel that the question isn't interesting to people and your observation is helpful.
$endgroup$
– kingW3
Dec 23 '18 at 16:36
add a comment |
1
$begingroup$
Not a complete answer which finds an expression for the $(x,y)$. However, here is an identity that I think will probably get you where you want to go. $max{{a,b}} = frac{a+b}{2} + frac{|a-b|}{2}$
$endgroup$
– Mason
Dec 23 '18 at 15:55
$begingroup$
I hope I didn't rob you of more complete answers by putting something more worthy as a comment as an answer. Anyway: I put a suggested edit and deleted my answer to get this back on the list of unanswered questions+ an edit will bump this back on to active questions.
$endgroup$
– Mason
Dec 23 '18 at 16:06
$begingroup$
@Mason Nice of you to do that, I don't think you robbed me of an answer. I feel that the question isn't interesting to people and your observation is helpful.
$endgroup$
– kingW3
Dec 23 '18 at 16:36
1
1
$begingroup$
Not a complete answer which finds an expression for the $(x,y)$. However, here is an identity that I think will probably get you where you want to go. $max{{a,b}} = frac{a+b}{2} + frac{|a-b|}{2}$
$endgroup$
– Mason
Dec 23 '18 at 15:55
$begingroup$
Not a complete answer which finds an expression for the $(x,y)$. However, here is an identity that I think will probably get you where you want to go. $max{{a,b}} = frac{a+b}{2} + frac{|a-b|}{2}$
$endgroup$
– Mason
Dec 23 '18 at 15:55
$begingroup$
I hope I didn't rob you of more complete answers by putting something more worthy as a comment as an answer. Anyway: I put a suggested edit and deleted my answer to get this back on the list of unanswered questions+ an edit will bump this back on to active questions.
$endgroup$
– Mason
Dec 23 '18 at 16:06
$begingroup$
I hope I didn't rob you of more complete answers by putting something more worthy as a comment as an answer. Anyway: I put a suggested edit and deleted my answer to get this back on the list of unanswered questions+ an edit will bump this back on to active questions.
$endgroup$
– Mason
Dec 23 '18 at 16:06
$begingroup$
@Mason Nice of you to do that, I don't think you robbed me of an answer. I feel that the question isn't interesting to people and your observation is helpful.
$endgroup$
– kingW3
Dec 23 '18 at 16:36
$begingroup$
@Mason Nice of you to do that, I don't think you robbed me of an answer. I feel that the question isn't interesting to people and your observation is helpful.
$endgroup$
– kingW3
Dec 23 '18 at 16:36
add a comment |
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1
$begingroup$
Not a complete answer which finds an expression for the $(x,y)$. However, here is an identity that I think will probably get you where you want to go. $max{{a,b}} = frac{a+b}{2} + frac{|a-b|}{2}$
$endgroup$
– Mason
Dec 23 '18 at 15:55
$begingroup$
I hope I didn't rob you of more complete answers by putting something more worthy as a comment as an answer. Anyway: I put a suggested edit and deleted my answer to get this back on the list of unanswered questions+ an edit will bump this back on to active questions.
$endgroup$
– Mason
Dec 23 '18 at 16:06
$begingroup$
@Mason Nice of you to do that, I don't think you robbed me of an answer. I feel that the question isn't interesting to people and your observation is helpful.
$endgroup$
– kingW3
Dec 23 '18 at 16:36