I am trying to solve a QCLP problem of the type \begin{equation*} \begin{aligned} & \underset{x,y}{\text{max}} & & \sum_xx_iU_i + \sum_yy_iVi \\ & \text{subject to} & & \frac{\sum_ix_ia_i^jb_i^j}{\sum_ix_ia_i^j} - \frac{\sum_iy_ic_i^jd_i^j}{\sum_iy_ic_i^j} \in [-A^j,A^j], j = 1, \ldots, n\\ & & & x_i \geq 0, \forall i \\ & & & y_i \geq 0, \forall i \\ & & & \sum_ix_i=1 \\ & & & \sum_iy_i=1, \\ & & & \textit{...some other linear constraints...}, \end{aligned} \end{equation*} My issue is with the $2n$ quadratic inequalities. All the $a_i,b_i,c_i,d_i$ are nonnegative. The first approach I had was to re-arrange the constraints in a form $w^TM^jw<0$, where $w$ is a vector that contains x and y $\textit{stacked}$ together. The matrix $M^j$ is a symmetric matrix with $0$ on the diagonal (actually a block of $0$ on the top left and bottom right corner). This matrix is not semi-definite positive. Consequently, the constraint is not convex.
All my attempts to re-arrange/apply transformations have been unsuccessful so far. I was wondering if it would be possible to find some $\textit{hidden}$ convexity using the fact that all the $x_i, y_i$ are nonnegative (and maybe also the fact that the intervals are of the form $[-A,A]$)?
Thank you.