I'm trying to solve the following minimization problem: $$ x = \arg\min_x f(x) = \arg\min_x \sum_i (x^\top A_i x - b_i)^2,\\[0.5em] x\in\mathbb{R}^n, \quad b_i\in\mathbb{R}^{+},\\[0.5em] \quad A_i\in \mathbb{R}^{n\times n},\,\, A_i \text{ positive definite and symmetric} $$
After some simulations, there seems to be several local minima, with an error $f(x)>0$. For instance, with $n=2$ and for a set of valid values of $A_i$ and $y_i$, we have 4 local minima:
On the other hand, I believe this problem is equivalent to finding $x$ that gives the least-squares error of the system of equations with constraints $x^\top A_i x = b_i$ (not sure if this is useful, though).
Can the structure that this problem be leveraged to reach a non-iterative solution to it?
Any help is highly appreciated.