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 $3$).
You are given a vector $a \in \mathbb R^n$, a direction $v \in \mathbb R_n \backslash \{ 0 \}$, and a point $p \in \mathbb R^n$. Write $$ (p-a) = \left( \frac{(p-a) \cdot v}{v \cdot v} \right) v + \left( (p-a) - \left( \frac{(p-a) \cdot v}{v \cdot v} \right) v \right). $$ The two vectors in the sum are orthogonal, the factor $\eta$ you are looking for is $\left( \frac{(p-a) \cdot v }{v \cdot v} \right)$ and the norm of the vector on the right of the sum is the minimal distance you can acheive.
All of this is elementary linear algebra : I projected the vector $p-a$ along the line passing through $a$ in the direction of $v \neq 0$, and the scaling factor of the length of $v$ versus the length of the projection is the factor $\eta$. If you compute $p-a$ minus the projection vector you get the vector that starts at the minimal point on the line and goes to the point, so computing its norm gives you the minimal distance.
Hope that helps,