I have a set of $n$ linear equations of the form:
$x_n = \phi_n + v\left(w_n - q_n\right),\text{for}\quad q_n \le w_n$ and
$x_n = \phi_n ,\text{for}\quad q_n > w_n$.
I am looking for guidance on how to minmize the function:
$ \underset{v}{\text{minimize}} \quad g= \sum_{i=1}^{n} \left| 4m\pi + \frac{14}{6} \pi - x_i\right| \text{for}\quad v >0$,
where m, $\phi$, $w_n$, $q_n$ are all constants.
Approach: I am thinking of optimiszing $g^2$ instead and attempting to use a gradient descent method. I am not sure if this is possible though as I am only interested in optimizing in one direction in terms of $v$? Any guidance/help/advice would be greatly appreciated!
If all symbols other than $v$ and $x$ are parameters, rather than variables, I would consider using Golden Section search or bisection search (on $g$, no squaring needed).