I have this optimization problem that requires finding $x_i$s satisfying following criteria:
$$ \min_{x}\ \sum_{i=1}^n\left( \frac{|x_i|}{y_i}\right) $$ such that, $$ u > y_i - x_i > l, $$ $$ \sum_{i=1}^n(y_i - x_i) = k $$
Here $u, l, k$ and $y_i$ are known. Is it possible to solve it analytically? If not then does replacing $\sum\limits_{i=1}^n\left(\frac{|x_i|}{y_i}\right)$ by $\sum\limits_{i=1}^n\left(\frac{x_i}{y_i}\right)^2$ helps?
Any suggestion is welcome.
Thanks!