Let $X,Y\in\mathbb{R}^{n\times k}$, $\Lambda(\alpha) = \text{diag}(\alpha)$, with $\alpha\in\mathbb{R}^k$, and let $c,d\in\mathbb{R}^+$ be positive constants. Let

$$A_i(\alpha) = (X\Lambda(\alpha) X^T)^{-1}x_ix_i^T(X\Lambda(\alpha) X^T)^{-1},$$ $$B_i(\alpha) = (Y\Lambda(\alpha) Y^T)^{-1}y_iy_i^T(Y\Lambda(\alpha) Y^T)^{-1},$$ where $x_i, y_i\in\mathbb{R}^n$ are the $i$-th column of matrix $X$ and $Y$ respectively.

Is there any efficient way to solve the following system of equations for $\alpha$?

$$ c_i(x_i^TA_i(\alpha)x_i) = d_i(y_i^TB_i(\alpha)y_i), \quad i = 1,\dots k. $$