For $A\succ 0, B\succeq 0, \rho,\phi>0$, and a given $y$, I am trying the find a cheap yet numerically stable method for solving a sequence of the following problem (it is a penalty method and $\rho$ gradually goes to infinity):
\begin{equation} \min_{x\in \mathbb{R}^n} x^TBx+\rho\|x-y\|^2 \qquad \text{ subject to } \qquad x^TAx\ge \phi. \end{equation}
The SDP relaxation of this problem is exact and the solution of the relaxation leads to a solution of the original problem above (even if it is not ranked one). But the SDP relaxation is not stable in my numerical experiments (I am using CVXPY and when $\rho$ becomes large, it gives me $-\infty$ as the optimal value, even though clearly my problem is bounded below.)! So, I have to replace it with another effective method. The ideal case is that the method is cheap (so perhaps avoids decomposing the Hessian of the objective function) and stable such that it can be applied to large-scale problems as well.