I'd like to solve the following quadratically constrained quadratic program (QCQP)
\begin{equation}\label{bijective} \begin{split} \min_{x} \quad &x^{T}Ax\\ \mathrm{s.t.}\quad &c_k(x)\leq0,\quad k=1,2,\ldots,N \end{split} \end{equation}
where $A$ is a sparse positive semidefinite matrix, $c_k(x)$ are non-convex quadratic functions with about 40000 variables. The Jacobian matrix $J=[\nabla c_1(x),\nabla c_2(x),\ldots,\nabla c_{N}(x)]$ is also sparse.
I use the MATLAB function fmincon to solve this problem and choose the sqp algorithm, but it is very slow and can't get a local minimum solution in a limited time.
How to solve the above problem efficiently? Is are any efficient software to do this?