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?

  • 1
    $\begingroup$ 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
  • 2
    $\begingroup$ 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

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.