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.