# Algorithm for solving system of quadratic equations and linear equations

Let $x \in R^N$. From a Spectral Chebyshev collocation method, I have a system of quadratic and linear equations. Denote them, $$x^T Q_i x + L_i^T x = 0$$ and $$A x = 0$$ Furthermore, I know the solution will fulfill $$B x \geq 0$$ for $Q_i \in R^{N \times N}, L_i \in R^{N}, A\in R^{M \times N}, B\in R^{S\times N}$, and $i = 1,\ldots P$. To get a sense of sizes, I have $N \approx M\approx S \approx P \approx 400$. The matrices $A$ and $B$ would have about $N/2$ nonzeros and each $Q_i$ is going to have around $N/4$ nonzeros.

Question: I can solve this with a nonlinear solver, but are there any other methods (and accompanying software implementations) which can solve this and exploit the fact that everything is linear or quadratic? I have no reason to believe that $Q_i$ lead to any particular structure (e.g., not necessarily positive semi-definite, etc.). That said, I am willing to throw caution to the wind and accept local solutions (especially if there is appropriate software I can throw the problem at).

Apologies for my general ignorance of these types of problems. I have looked at QCQP methods, but they seem to focus on inequality constraints. I am also not sure I can naively put the quadratic constraints as objectives to minimize due to their unknown structure/convexity.

Any polynomial equation can be re-written as a system of quadratic equations. (For example: $x^3 = 1$ is equivalent to $\{xy = 1,\, x^2 - y = 0\}$.) So solving a set of quadratic equations is not fundamentally easier than solving a set of polynomial equations.
The advantage of these packages, compared to a non-linear solver is that you would be guaranteed a solution if one exists. The disadvantage is that the search tree grows exponentially in the number of unknowns, so $N \approx 400$ might be impossible unless the inequalities permit very efficient pruning.