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I have a problem of allocating x between p entities, each entity has a constant $\rho_i$ such that the allocated part should be proportionnal.

Without loss of generality we can divide the ressources by x and normalise the $\rho_i$ by $\Sigma_i\rho_i$ to see the unconstrained problem has a simple solution:

$$x_i = n \frac{\rho_i}{\Sigma_i\rho_i} $$

My problem is to solve a constrained problem, such that:

$$ a_i \leq x_i \leq b_i $$

I have an iterative algorithm, starting at $x_i = n \frac{\rho_i}{\Sigma_i\rho_i} $ at each iteration we reallocate the ressources that excess the constraint. Theoretically this works because $\Sigma b_i > 1$ and $\Sigma a_i < 1$.

I have the feeling that it could be reformulated as an optimization problem but can't figure out what the objective function would be.

Lucas Morin
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    You could try to minimize $\sum (x_i - y_i)^2$ subject to $a_i \le x_i \le b_i$ where $y_i = n \rho_i / \sum_i \rho_i$ (in English, compute the assignment that satisfies the constraints and that least deviates from the assignment you would have if there were no constraints) – BrunoLevy Nov 14 '15 at 14:38
  • I think the reallocation a each step is made proportionnaly to $\rho_i$, so i think the errors should take into account $\rho_i$ in some way. – Lucas Morin Nov 14 '15 at 16:07
  • @Were_cat: I'm pretty sure Bruno is suggesting $y_j = \rho_j /\sum_i \rho_i$ be used to minimize least square error $\sum (x_i - y_i)^2$. – hardmath Nov 14 '15 at 19:46
  • let say I have three entity, one does not verify the constraint b, so the amount will be corrected by x1-b on the first entity, so I have to reallocate x1-b on the others entities. I am under impression that the reallocation will be done equally between other entities, not proportionaly to their their natural proportion $\rho_i$. – Lucas Morin Nov 15 '15 at 01:08

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