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I have four numbers $X_1$, $X_2$, $X_3$ and $X_4$ that range from [-2,2]. I want to scale these numbers ($x_1=scaled(X_1)$) with the following constraints:

$$x_1+x_2+x_3+x_4=1,$$ subject to $$x_1+x_2<1,$$ $$x_2+x_4<1,$$ $$x_3+x_4<1,$$ $$x_1+x_3<1.$$ The $x_i$ can be between $[-\infty,\infty]$, as long as the constraints above are satisfied. How can this be achieved?

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    $\begingroup$ Why? What are you trying to achieve by that? You could replace each value with 0.25 and the constraints would hold, but I guess there are also some other assumptions that you are making that you didn't mention. $\endgroup$ – Tim Apr 13 at 11:37
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    $\begingroup$ Does "scaling" mean applying a (nontrivial, meaning that the multiplier is nonzero) linear transformation? $\endgroup$ – Lewian Apr 13 at 11:42
  • $\begingroup$ Im trying to achieve weights that sum to 100% and the subweights also sum to 100%. The X_i's are Betas in a regression and I want to scale these Betas into weights, that sum to 1, and has the constraints listed above. $\endgroup$ – niclas englesson Apr 13 at 12:07
  • $\begingroup$ What for? This sounds like an XY problem. If those are parameters of the model, why not just use constrained optimization with those constraints? $\endgroup$ – Tim Apr 15 at 6:57
  • $\begingroup$ Because I need a formula so that I can do this for rolling 24month periods, of course for a one time solution it would be easier to add constraints to the optimization, but I'm trying to figure out a way to make it scalable (for thousands of regressions) and also easily updatable with new data. $\endgroup$ – niclas englesson Apr 15 at 13:44
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How do you want to treat negative individual values? Your constraints also require all pairs to add to values greater than $0$ as well as less than $1$, which suggests to me that you have some issues with negative weights.

You could try something like $$\mathrm{scaled}(X_i) = \frac{X_i-\min\{X_j,0\}}{\sum\limits_k\left(X_k-\min\{X_j,0\}\right)}$$ on some examples of your choosing. It requires the $X_i$ not to be all equal to some non-positive value, and if they were all equal then $\mathrm{scaled}(X_i)=\frac14$ would be the obvious choice

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