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I've got to solve an optimisation problem and don't know how to approach this (I'm thinking of writing a script to solve the problem recursively but wonder if there's another approach).

The target is given by $n$ real numbers, e. g. $a_{1t} = 10$, $a_{2t} = 20$, $a_{3t}=30$.

I've got the current value of the $n$ numbers, e. g. $a_{11}=1$, $a_{21}=1$, $a_{31}=5$.

I can change these numbers as follows:

$a_{12} = a_{11} + \eta · \mu_1 $

$a_{22} = a_{21} + \eta · \mu_2 $

$a_{32} = a_{31} + \eta · \mu_3 $

I need to find $\eta$ so that the mean square root between the two sequences is minimal when changing the $n$ numbers according to this rules where $\mu_1$, $\mu_2$ and $\mu_3$ are fixed factors (e. g. $1$, $2$ and $3$).

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    If $\mu_1, \mu_2, \mu_3$ do not depend on $t$, your problem is way more simple ; iterating will only "change the value of $\eta$ in the first iteration". Do you see what I mean? For instance, $$a_{13} = a_{12} + \eta_2 (\mu_1) = (a_{11} + \eta_1(\mu_1)) + \eta_2 (\mu_1) = a_{11} + (\eta_1 + \eta_2) \mu_1,$$ so that anything you could do in $t$ iterations you can do in one.2012-04-06
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    Yes, $\mu_1, \mu_2, \mu_3$ do not depend on $t$.2012-04-06
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    Your problem only has two steps then, the beginning step and the least square solution. Adding multiple steps in between is meaningless, unless you're trying to use a numerical algorithm to converge to the solution, but in this case analytical solutions exist. Would that be what you would be looking for?2012-04-06
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    I guess it is. My problem is I don't exactly know how to "understand" this problem. I also thought about taking it as a vector problem - a n-dimensional target vector is given and another vector is given which should be altered to be closer to the target vector, but only certain components following certain rules (depending on the $\mu$s) can be altered.2012-04-06
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    If what you're saying means your problem is a one-step problem, then you know that there is a solution to the minimal distance between a line and a point in $\mathbb R^n$, do you?2012-04-06
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    No I don't. Probably this is what I'm looking for..2012-04-07
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    Yeah, I think I'm looking for the minimum distance between a line and a point in $\mathbb{R}^n$. But I need to know the corresponding value of $\eta$ as well not only the value of the minimum distance.2012-04-07

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