I want to find a (local) minimizer $x, y$ to the following optimization problem:
$ \min_{x,y}\ x^T x + a^T x \qquad \mathrm{s.t.} \qquad \begin{array}{r}x^T M_i y + c_i = 0 \\ y \geq 0\end{array}.$
Is sequential quadratic programming my only recourse, or are there specialized techniques that might be more efficient? Although the problem is clearly non-convex, the special features (the constraint is quadratic, and the inequality-constrained variables appear only linearly in the constraints and not at all in the objective) give me some hope.