Given an inequality through a quadratic polynomial

$$ ax_1^2+bx_2^2 +cx_1x_3+dx_2x_4\leq 0,$$

where $c_i\in \mathrm{R}$, $0\leq x_i \leq 1$, $i=1,2,3,4$, can it be somehow transformed into an inequality in terms of $y_i$, $i=1,2,3,4$, where

$$ \sum_{i=1}^4 x_i^1 = y_1, \sum_{i=1}^4 x_i^2 = y_2, \sum_{i=1}^4 x_i^3 = y_3, \sum_{i=1}^4 x_i^4 = y_4$$

establishes the relation between $x_i$ and $y_i$.

How does on go from a function $f(x_1,x_2,x_3,x_4)$ over to a function $g(y_1,y_2,y_3,y_4)$? Is such a transformation possible? If so, in what direction proceed to prove its existence? How to explicitly construct such a transformation?

I am not even sure how to start, any hints on improving this question welcome.

  • $\begingroup$ I can't see how powers of 3 or 4 can play any role. You also have not written down any inequalities involving your sums so you may need to clarify your question.. $\endgroup$ – Paul Oct 4 '18 at 14:14
  • $\begingroup$ @Paul The powers of 3 and 4 define $y_3$ and $y_4$. We have 4 equations of 4 variables which gives us the transformation. $\endgroup$ – wondering Oct 4 '18 at 14:46
  • $\begingroup$ @wondering The $y_j$ are symmetric in $x_i$ while the inequality is not. So short answer is no, there is no such transformation (unless of course you have additional conditions). $\endgroup$ – dxiv Oct 4 '18 at 16:55

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