I have have a vector of weights, $y$ (summing to 1 and each greater than 0). However, these weights violate some inequality and equality constraints. Thus, I would like to adjust these weights, so that they don't violate the constraints.
I initially thought of minimizing the squared deviations between the original weights and the new ones subject to the constraints. Something along the lines of:
$\min_x \sum (x_i-y_i)^2$
s.t.
$Ax \leq b$
$Cx = d$
However, I would like to "weight" my solution with a signal, $W$, such that the $x$'s with a high signal, get a higher value.
I initially thought of doing a weighted least squares of the form
$\min_x \sum W_i (x_i-y_i)^2$,
but (as I see it) this weighting scheme will force $x_i$'s with a large signal ($W_i$) closer to the original weight, $y_i$, and not increase the value of these $x_i$'s.
Furthermore, weighting with the inverse of the signal ($1/W_i$), lets $x_i$'s with high signals deviate further from the original weight, but the deviations could be in both directions.
How should I pose my problem, so that $x$'s with a high signal, get a higher value?